# Peduncle Gripping and Cutting Force for Strawberry Harvesting Robotic End-
effector Design
††thanks: This work was supported by Lincoln Agri-Robotics as part of the
Expanding Excellence in England (E3) Programme.

Vishnu Rajendran S1,3, Soran Parsa2,3, Simon Parsons2,3, Amir Ghalamzan
Esfahani1,3 Lincoln Institute of Agri-Technology (LIAT)1, School of Computer
Science2, Lincoln Center of Autonomous Systems (LCAS)3
University of Lincoln
Lincoln, United Kingdom

###### Abstract

Robotic harvesting of strawberries has gained much interest in the recent
past. Although there are many innovations, they haven’t yet reached a level
that is comparable to an expert human picker. The end effector unit plays a
major role in defining the efficiency of such a robotic harvesting system.
Even though there are reports on various end effectors for strawberry
harvesting, but there they lack a picture of certain parameters that the
researchers can rely upon to develop new end effectors. These parameters
include the limit of gripping force that can be applied on the peduncle for
effective gripping, the force required to cut the strawberry peduncle, etc.
These estimations would be helpful in the design cycle of the end effectors
that target to grip and cut the strawberry peduncle during the harvesting
action. This paper studies the estimation and analysis of these parameters
experimentally. It has been estimated that the peduncle gripping force can be
limited to 10 N. This enables an end effector to grip a strawberry of mass up
to 50 grams with a manipulation acceleration of 50 m/s2 without squeezing the
peduncle. The study on peduncle cutting force reveals that a force of 15 N is
sufficient to cut strawberry peduncle using a blade with a wedge angle of
16.60 at 300 orientation.

###### Index Terms:

selective harvesting robots, end effectors, strawberries, peduncle

##  I Introduction

Strawberries are one among the high-yield crops that have received great
attention from researchers in developing robotic solutions for harvesting.
Robotic harvesting of strawberries has been predominantly researched and
developed for those which are grown in greenhouse scenarios[5]. A selective
harvesting method is primarily utilized to harvest the strawberries because
all fruits on the plant may not be ripe at the same time. And moreover, it
needs some judgment to pick the ripe fruit which may be staying in isolated
condition or as partially/completely occluded by adjoining fruits or by the
parent plant. The clustered nature of strawberries makes it troublesome for
any robotic intervention to reach an efficiency competent to a human picker.
Several strawberry harvesting robots are reported in the literature [9, 8, 7,
2, 15, 16, 3, 11].

In these robots, the end effector (EE) is the unit that directly and
physically interacts with crops. Across different EE technologies for
strawberry harvesting, these physical interactions include: (i)
gripping/grasping the strawberry by its peduncle or fruit body (attachment),
(ii) detaching from the plant by pulling, twisting the fruit, or cutting the
peduncle, (iii) facilitating the fruit transport from the detachment location
to the storage, and (iv) pushing/parting of strawberries in the cluster for
obstacle separation during detaching action [14] which is a recently explored
functionality.

Figure 1: Strawberry harvesting end effectors: [a]. Parallel jaw mechanism for
simultaneous attachment and detachment[9], [b].Suction head for attachment and
blade action for detachment[2], [c]. Suction head and two jaw fingers for
attachment and bending action of end effector for detachment[16], [d].Suction
cup for attachment and thermal based detachment[7], [e]. Soft gripping for
attachment and followed by rotation with pulling action for detachment[4],
[f]. Detachment without any attachment[13]

Among attachment and detachment actions, some EEs performs simultaneous
gripping and detachment of the strawberry peduncle using a parallel jaw
mechanism [9, 10] (figure 1a). In such a mechanism, one jaw will be shaped in
the form of a cutting blade or is provided with a provision to attach
replaceable blades. One such EE design makes use of a suction cup to provide
an additional grip by sucking the fruit body to avoid any positional errors
during this simultaneous gripping and cutting actions [9]. A suction-based
approach is used by another end effector which uses a suction head to grip the
fruit body and then rotates so that a blade arranged on the curved opening of
the suction head trims the peduncle [2] (figure 1b). Instead of using any
blades for cutting the peduncle, the EE reported by Yamamoto S et al. [16]
uses a bending action to detach the strawberry after gripping the fruit body
by a suction head and a two-jaw gripper (figure 1c). Also, a thermal-based
cutting is reported for another EE that uses an electrically heated wire on
the gripping jaw to cut the peduncle [7]. In this EE, once the fruit body is
gripped by the suction cup to position the peduncle between the two jaws
(cutting device), the jaws then close and cuts the peduncle using the heated
wire (figure 1d). The end effector developed by Octinion uses a soft gripper
to grip the strawberry fruit body and imposes a rotational motion while
pulling the strawberry to detach it from the peduncle [4] (figure 1e). The EEs
reported above make either a gripping contact with the fruit body or with the
peduncle during the harvesting action. But the EE reported by Xiong Y et al.
[13] doesn’t grip the fruit body or the peduncle during the harvesting action.
Instead, a combination of three active and passive fingers guides the
strawberry into the EE housing. Once the fruit reaches the cutting location,
scissor-shaped blades cut the peduncle to detach the strawberry (figure 1f).

Considering the different options for gripping and cutting, it is always
beneficial to avoid applying any forces on the fruit body by the EE contact
surfaces. Since strawberry is a very soft and delicate fruit, there are higher
chances of bruising during such operations. Aliasgarian et al. [1] showed that
the strawberry fruits are more damaged when exposed to compression forces on
their body. Hence, from the EE design point of view, it is recommended to
target the peduncle for gripping/cutting actions or to avoid any grip action
as demonstrated by Xiong Y et al. [13].

To develop an end effector solution that attempts to detach strawberries by
cutting the peduncle, it is essential to understand certain design parameters.
This includes the estimation of the required cutting and the gripping force
that can be applied on the peduncle. Knowing the limit of gripping force
eliminates the chances of a loose grip or crushing the peduncle during the
gripping action. These situations can lead the detached strawberry to fall off
from the grip while manipulating it during the harvesting cycle. Moreover,
knowing these force gives an insight for the selection of actuation system for
the EE.

The only study that we are aware of which reports the measurement of the force
required to detach the peduncle from a strawberry is by Dimeas F et al. [6].
This considers the use of a pulling effort (technique A) and a combination
bending-pulling effort (technique B). Technique A needed a force of 13.94 N
for a peduncle with mean diameter of 2.17 mm, while technique B took 3.17 N to
detach a peduncle with a mean diameter of 1.78 mm. Moreover, Dimeas F et al.
[6] also reported that a safe gripping force below 10 N can be applied on the
strawberry fruit body for detaching the peduncle by technique B such that it
won’t bruise the fruit body. Apart from these measured forces, no work has
been reported which studied the cutting force and the limit of gripping force
that can be applied on the strawberry peduncle. Given this lack of data, this
paper studies, the cutting and gripping force that must applied to the
strawberry peduncle, helping future work on EEs targeting peduncle for the
selective harvesting of strawberries.

For presenting the study details, this paper is organized in the following
order. Section II outline the experimental studies conducted, and section III
discusses the results obtained. The conclusive remarks of the paper are
presented in section IV followed by study limitations in section V.

##  II Experimental Study

For these experimental studies, we have used two varieties of strawberries
namely, Katrina and Zara grown in the polytunnel at Riseholme Campus of the
University of Lincoln, UK. The peduncle specimens were collected from both
varieties in such a way that there will be uniformity in their diameter ranges
for better results comparison.

###  II-A Study on the gripping force limit

This study intended to estimate the limit of the gripping force that can be
applied on the strawberry peduncle without crushing it. To understand this
force limit, experiments were conducted by applying compression force
(analogous to the gripping force) to the peduncle specimens using a Universal
Testing Machine (UTM). The peduncles of ripe strawberries of both varieties
were selected for preparing the specimens. 15 specimens of each variety were
prepared so that the peduncle was 10 mm in length and were trimmed at a
distance of 10 mm from the top surface of the ripe strawberry fruit. This
specimen measurement can simulate the situation where a two jaw parallel end
effector grips the peduncle within 10-20 mm from the top surface of the
strawberry top surface during harvesting. The specimen diameter varied from
1.40 mm to 2.22 mm for Katrina with a mean and standard deviation of 1.75 mm
and 0.24 mm. And for Zara, the diameter varied from 1.43 mm to 2.33 mm with a
mean and standard deviation of 1.76 mm and 0.25mm.

During the experiment, each specimen was placed on the base plate of UTM as
shown in the figure 2. The UTM was configured in such a way that the vertical
ram (with load cell attachment) approached the base plate at a rate of 1.5
mm/min. Once the ram starts touching the specimen, the connected PC records
the interaction force at a resolution of 0.0001 N. The force values recorded
during the study are presented in section III.

Figure 2: Universal testing machine (UTM) for applying and recording the
compression force on specimens used for our experimental study.

###  II-B Study on cutting force

This study aimed to estimate the force required to cut the peduncle of a ripe
strawberry using an off-the-shelf blade profile. As an extension to this, the
variation of this force at different cutting orientations was also studied.
The profile of the selected cutting blade was studied using a scanning
electron microscope. The blade had a double bevel cutting edge with a wedge
angle of 16.60 and a thickness of 0.22 mm (figure 3).

Figure 3: Blade profile measured using Scanning Electron Microscope

As shown in the figure 4a, we designed and 3D printed some fixtures in order
to conduct the experiment. These include a blade holder for holding the blade,
a peduncle support on which to place the peduncle specimen, and a pair of
press-fit clips for positioning the specimen on the support. Also, angular
wedges of 100, 200, 300 were made that can be used to tilt the peduncle
support loaded with the specimen. This allowed us to study the cutting force
variation for 00,100, 200, 300 orientations. The same UTM used in the previous
study was used here as well. The blade holder was attached to the vertical ram
of UTM using a thin layer of adhesive and the peduncle support to the base
plate as shown in the figure 4b-c. 15 peduncle samples from both Zara and
Katrina varieties were prepared for this study, i.e., 15 samples from each
variety for each orientation of cut. These specimens were prepared by cutting
the peduncle at a length 30 mm from the top surface of the ripe strawberry.The
diameter range of the specimens used for each trails are shown in the table I.

TABLE I: Diameter of the peduncle specimens used to study the cutting force
Specimen Diameter (mm)
---
Trials | Zara | Katrina
Minimum | Maximum | Minimum | Maximum
00 | 1.03 | 2.14 | 1.10 | 2.00
Mean(SD):1.53(0.31) | Mean(SD): 1.51(0.29)
100 | 1.20 | 2.10 | 1.00 | 2.00
Mean(SD): 1.59(0.30) | Mean(SD):1.54(0.27)
200 | 1.11 | 2.20 | 1.10 | 2.13
Mean(SD): 1.54(0.31) | Mean(SD): 1.58(0.32)
300 | 1.12 | 2.10 | 1.00 | 2.00
Mean(SD): 1.52(0.28) | Mean(SD): 1.54(0.30)

Figure 4: (a). 3d printed fixtures (b). Experimental setup of studying cutting
forces and its variation with change in angle of cut

During the trails, the cutting blade moved along with the vertical ram of the
UTM (at the rate of 1.5 mm/min) pierced the specimen center, and cut the
specimen peduncle into two halves. This simulates the peduncle cutting
happening at 15 mm from the top surface of strawberry using a parallel cutting
mechanism. While the blade was interacting with the stem, the load cell values
were recorded by the connected PC and were used to identify the peak force
required for cutting each specimen. The force recordings are presented in
section III.

##  III Results and Discussion

###  III-A Gripping force study

During the experimental trials, all tested specimens showed a common force
profile under compression load which is shown in the figure 5. While applying
compression load (by the extending UTM ram) to the specimen, there was a
gradual increase in the resistive force to a certain point, from then it
showed a sudden drop. After then, the specimen got squeezed completely on
further application of compression load. It has been noticed that, after the
drop in the resistive force, the specimen went to permanent deformation before
getting completely squeezed. This peak force before the drop (FC), is
considered as the point of interest and is recorded for all specimens against
their respective diameter (figure 6). The trend of this force (FC) can be
studied to limit the gripping force on the peduncle such that there is a
lesser chance of squeezing the peduncle during the gripping action.

Figure 5: Force profile recorded for the specimens while applying compression
load by the extending UTM ram. The final squeezing force is not presented here
as the peak force (FC) before sudden drop is the point of interest

Figure 6: Variation of FC with respect to the specimen diameter

Figure 7: Frequency distribution of FC measured for all specimens

Considering the peak forces, presented in the figure 6, we can determine that
the lowest of these peak forces (FC) recorded is about 26.83 N and 22.53 N for
Katrina and Zara respectively. This means that at these lowest values of
compression force (analogous to gripping force), the respective test specimen
went into permanent deformation before squeezing. Hence if we allocate a
factor of safety of 2 to the lowest of these two values of forces (26.83 N and
22.53 N), the gripping force should be limited to around 10 N. Assuming that
for a two jaw parallel gripper, if we use a soft material lining on the
gripping surface with a coefficient of friction of 0.3, then under static
condition (see equation 1, a = 0 m/s2), the end effector will be able to
handle a mass of 300 grams. Assuming that the mass of the strawberry to be
gripped in the end effector is about 50 grams and keeping other parameters the
same, under dynamic conditions, the end effector would be able to handle this
50 gram strawberry for a manipulator acceleration of up to 50 m/s2.

$F_{g}=\frac{m.(g+a).S}{\mu}$ (1)

where; ’Fg’ is the net gripping force (N), ’m’ is the mass to be handled (Kg),
’g’ is the acceleration due to gravity (m/s2), ’$\mu$’ is the co-efficient of
friction, and ’S’ is factor of safety[12].

This computed load carrying capability at 10 N is more than enough for an end
effector to handle a ripe strawberry in static and dynamic conditions of a
harvesting cycle. It is to be noted that, in dynamic conditions, for the same
gripping force, the loading capacity can be increased by regulating the
manipulator acceleration.

The frequency distribution of the peak forces (FC) of the tested specimens is
shown in figure 7. The majority of the forces (FC) fall in the range of 31.00
N to 40.99 N for both the varieties, when the diameter of the specimens shares
a close mean of 1.75 mm and 1.76 mm (with a standard deviation of 0.24 and
0.25) for Katrina and Zara respectively. Since the computed
compression/gripping force limit (10 N) is well below this range, it can be
concluded that 10 N would be a safe force that can be applied to the peduncle
for an effective gripping.

Figure 8: Force profile recorded for the specimens during cutting test

Figure 9: Variation of FP with respect to specimen diameter at test cutting
orientations (Katrina)

Figure 10: Variation of FP with respect to specimen diameter at test cutting
orientations (Zara)

###  III-B Cutting force study

While analyzing the force values recorded during the cutting trials, again a
common force profile has been noticed for all the tested specimens. A sample
force profile is shown in the figure 8. In the profile, there is an increase
in force value during the cutting action but with two sudden drops after two
force peaks (FP1 and FP2). After the second peak force, there is a flat
profile and followed by a sharp rise in force after a point (E). This sharp
increase happens when the blade touches the peduncle supports after cutting
the peduncle off. So from the force profile for each specimen, the maximum of
the two peak forces (FP1 or FP2) is taken as the peak cutting force (FP)
required for that specimen. The peak cutting forces recorded for all specimens
against their diameter and at different orientations are shown in the figure 9
and 10.

From the force values recorded at different cutting orientations, it has been
noticed that the mean cutting force shows a relative lowest value at 300
orientation compared to other studied orientations. These mean cutting force
values (FPM) with respect to the mean diameter (DM) of the tested specimens at
different orientations are presented in table II. Also, it is seen that at 100
orientation, the mean cutting force (FPM) shows a relatively higher value.
These variations are found to be true for both the strawberry varieties
studied.

TABLE II: Variation of FPM with DM of specimens at test cutting orientations. Standard deviations are presented in brackets Orientation | DM (mm) | FPM (N) | DM (mm) | FPM (N)
---|---|---|---|---
of cut | [Zara] | [Zara] | [Katrina] | [Katrina]
00 | 1.53(0.31) | 4.95(1.33) | 1.51(0.29) | 6.71(2.05)
100 | 1.59(0.30) | 6.22(2.03) | 1.54(0.27) | 7.17(1.36)
200 | 1.54(0.31) | 5.11(1.27) | 1.58(0.32) | 5.08(0.98)
300 | 1.52(0.28) | 4.13(0.90) | 1.54(0.30) | 5.00(1.40)

At 300 cutting orientation, the maximum of the peak cutting force (FP)
recorded is about 7.20 N for Katrina, and 5.80 N for Zara. And hence, with a
factor of safety 2, the cutting force requirement can be approximated to 15 N
which could be considered sufficient to cut the strawberry peduncle at 300
orientation. Also, this force would be sufficient to handle other cutting
orientations studied. It is to be noted that this cutting force is measured at
a blade speed of 1.5 mm/min, so at higher blade speed, the required force will
be lesser than 15 N.

##  IV Conclusions

This work reports the studies conducted to estimate a few key parameters which
can help in the design of strawberry harvesting EEs targeting the peduncle.
The studied parameters include, the limit of gripping force that can be
applied on the peduncle without crushing them, and the force required to cut
peduncle. These parameters are estimated for two varieties of strawberry
namely Katrina and Zara. The experimental studies have revealed that a safe
gripping force of 10 N can be applied to the strawberry peduncle. And this
gripping force is sufficient for a two jaw parallel gripper to retain a
strawberry with a mass of 50 grams at a manipulation acceleration up to 50
m/s2 and without squeezing the peduncle. With regards to the peduncle cutting
force, it is shown that a blade with a wedge angle of 16.60 can cut the
strawberry peduncle with a force of 15 N at an orientation of of 300. The
cutting forces at an orientation of 300 have been studied as slightly lowest
among the other cutting orientations of 00, 100, 200. This cutting force
estimation will be valid while using a similar standard blade profile in an
end effector with a parallel cutting mechanism.

We will make use of these parameters in designing a novel end effector for
strawberry harvesting, and we present them here in the hope that they will be
helpful to other researchers who are engaged in developing end effectors for
harvesting strawberries.

##  V Limitation of the study

This study has only used specimens from two varieties of strawberries, and it
is an untested assumption that the estimated figures and trends can be a
representation of other varieties of strawberries within close tolerances.
Since our proposed end-effector design calls only for blade orientation below
300, we have not considered angles above 300 in the cutting force study. Hence
the cutting force trend for greater blade orientations remains unknown.

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# An $\text{SL}(3,\mathbb{C})$-equivariant smooth compactification of rational
quartic plane curves

Kiryong Chung  Department of Mathematics Education, Kyungpook National
University, 80 Daehakro, Bukgu, Daegu 41566, Korea<EMAIL_ADDRESS>and
Jeong-Seop Kim  School of Mathematics, Korea Institute for Advanced Study, 85
Hoegiro, Dongdaemun-gu, Seoul 02455, Korea<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbf{R}_{d}$ be the space of stable sheaves $F$ which satisfy the
Hilbert polynomial $\chi(F(m))=dm+1$ and are supported on rational curves in
the projective plane $\mathbb{P}^{2}$. Then $\mathbf{R}_{1}$ (resp.
$\mathbf{R}_{2}$) is isomorphic to $\mathbf{R}_{1}\cong\mathbb{P}^{2}$ (resp.
$\mathbf{R}_{2}\cong\mathbb{P}^{5}$). Also it is very well-known that
$\mathbf{R}_{3}$ is isomorphic to a $\mathbb{P}^{6}$-bundle over
$\mathbb{P}^{2}$. In special $\mathbf{R}_{d}$ is smooth for $d\leq 3$. But for
$d\geq 4$ case, one can imagine that the space $\mathbf{R}_{d}$ is no more
smooth because of the complexity of boundary curves. In this paper, we obtain
an $\mathrm{SL}(3,\mathbb{C})$-equivariant smooth resolution of
$\mathbf{R}_{4}$ for $d=4$, which is a $\mathbb{P}^{5}$-bundle over the blow-
up of a Kronecker modules space.

###### Key words and phrases:

Rational quartic curves, Projective bundle, Equivariant Kodaira-Spencer map,
Elementary modification

###### 2020 Mathematics Subject Classification:

14E05, 14D20, 14N10

##  1\. Motivation and results

We work on the field $\mathbb{C}$ of complex numbers.

###  1.1. Rational curves and its moduli spaces

Two ways are well-known to compactify rational curves space. Let $X$ be a
smooth projective variety with a fixed embedding $X\subset\mathbb{P}^{r}$.

  * •

([FP97]) Let $\mathcal{M}_{d}(X)$ be the moduli space of _stable_ maps
$[f:C\rightarrow X]$ in $X$ with degree $\deg(f)=d$ and genus $g(C)=0$ .

  * •

([Sim94]) Let $\mathbf{M}_{d}(X)$ be the moduli space of _stable_ sheaves $F$
in $X$ with Hilbert polynomial $\chi(F(m))=dm+1$.

The first (resp. second) one $\mathcal{M}_{d}(X)$ (resp. $\mathbf{M}_{d}(X)$)
is often called by the Konstevich space (resp. Simpson space). The first
moduli space $\mathcal{M}_{d}(X)$ for a convex variety $X$ has at worst a
finite group quotient singularity (i.e., orbifold singularity) ([FP97, Theorem
2]). Both spaces play a crucial role in counting of the number of curves, as
they are known as Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and
also descendant invariants) respectively. Besides this aspect, it has played
an important role in birational geometry (specially, log minimal model program
or Fano/hyperkähler geometry), and the first-named author studied the
birational relationship between these spaces $\mathcal{M}_{d}(X)$ and
$\mathbf{M}_{d}(X)$, $d\leq 3$ in a various setting ([CK11, CHL18, Chu22]). To
analyze the geometric structure of the Simpson space $\mathbf{M}_{d}(X)$ in
higher dimensional variety $X$ (and degree $d$), it is necessary to study the
most basic case, namely, $X=\mathbb{P}^{2}$. For example, as shown in [FT04,
CCM16], the moduli space $\mathbf{M}_{d}(\mathbb{P}^{3})$, $d\leq 4$ consists
of (several) irreducible components such that the intersection part is
generically a fibration structure over the Grassimannian $\mathrm{Gr}(3,4)$ of
planes in $\mathbb{P}^{3}$. In the viewpoint of enumerative geometry, the
moduli space $\mathcal{M}_{d}(\mathbb{P}^{2})$ provides a solution of a long
standing problem called by _Konstevich’s formula_ ([FP97]). In detail, let
$N_{d}$ be the number of degree $d$ rational curves passing through $3d-1$
given points in general position in $\mathbb{P}^{2}$. In 1995, through the
work of Kontsevich ([Kon95]), it has been found a recursive formula to compute
$N_{d}$:

$N_{d}=\sum_{d_{1}+d_{2}=d}N_{d_{1}}N_{d_{2}}d_{1}^{2}d_{2}(d_{2}{3d-4\choose
3d_{1}-2}-d_{1}{3d-4\choose 3d_{1}-1}).$

Clearly $N_{1}=N_{2}=1$. Even the case of lower degrees has a very long
history. In the mid-1800s, Steiner (resp. Zeuthen) showed that $N_{3}=12$
(resp. $N_{4}=620$) ([Ste54, Zeu73]). Also $N_{5}$ was computed in middle of
1980.

On the other hand, prior to Kontsevich’s work, research on the geometry of
plane curves was focused on the study of Severi varieties $V_{d,\delta}$ (so
called, the space of curves with prescribed degree $d$ and the number $\delta$
of nodes), and J. Harris made significant contributions and provided direction
in this area. One of his notable achievements was proving the irreducibility
of the Severi variety $V_{d,\delta}$ ([Har86]). Related with this, in [DH88],
the authors mentioned the need for a smooth space with a moduli meaningful
points. As one may expect, the boundary points of plane curves space may have
bad singularity as possible and thus it is very hard to construct a smooth
moduli space globally.

Although the management of orbifold singularities of the Kontsevich space has
led to the development of a rich theory (so called, _perfect obstruction_
theory [BF97]) in enumerative geometry, the fundamental question about smooth
compactification of $V_{d,\delta}$ has been a central concern by following the
monumental work of Hironaka ([Hir64]). In this stream, we propose a smooth
compactification of the space of degree $4$ rational plane curves, which is
the first non-trivial case in $\mathbb{P}^{2}$.

###  1.2. Main results

As a continued work of [Chu21], we recall the result and unsolved part in the
view-point of _resolution of singularities_. Since a general plane rational
quartic curve $C$ has three nodal points $Z$ by degree-genus formula, we
consider space of pairs $(C,Z)$ in the relative Hilbert scheme of three points
over the complete linear system of degree $4$. Let us denote
$\mathbf{R}_{4}^{h}$ by the closure of the locus of the rational quartic
curves in this relative Hilbert scheme. Then there is a canonical projection
map $\mathbf{R}_{4}^{h}\rightarrow\mathbf{H}[3]$ into the Hilbert scheme of
three points on $\mathbb{P}^{2}$. On the hand, as identifying
$\mathbf{R}_{4}^{h}$ with the space of a stable pairs
$(s,\mathcal{E}xt^{1}(I_{Z|C}(1),\omega_{\mathbb{P}^{2}}):=F)$,
$s\in\mathrm{H}^{0}(F)$ and performing the wall-crossing in the sense of
variation of geometric invariant quotients ([Tha96, He98]), we obtain a smooth
contraction of the locus of rational quartic curves with three collinear
points. In fact, let $\mathbf{R}_{4}$ be the closure of the locus of rational
quartic curves in the Simpson space $\mathbf{M}_{4}(\mathbb{P}^{2})$, then we
have a commutative diagram as follows.

$\textstyle{\mathbf{R}_{4}^{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$sm.
blow-
down$\textstyle{\mathbf{R}_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{s}}$$\textstyle{\mathbf{H}[3]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathbf{K}}$

Here $\mathbf{K}$ is a _Kronecker modules space_ , which is a birational
contraction of $\mathbf{H}[3]$ along the locus $\widetilde{\mathbf{D}}_{3}$ of
collinear three points (For an explicit definition, see Section 2.1). Let
$\mathbf{D}_{3}$ be the image of $\widetilde{\mathbf{D}}_{3}$ along the map
$\pi:\mathbf{H}[3]\rightarrow\mathbf{K}$. The main result of the paper [Chu21]
is to describe the fiber of the map
$p_{s}:\mathbf{R}_{4}\rightarrow\mathbf{K}$ over $\mathbf{D}_{3}$. That is,
each fiber of $p_{s}$ over $\mathbf{D}_{3}$ parameterizes the stable sheaf
which is completely determined by its support (For detail, see (2) of
Proposition 2.8). It is remarkable that our proof is to use the elementary
modification of the direct image sheaf of the universal family of
$\mathcal{M}_{4}(\mathbb{P}^{2})$. From this, we can conclude that $p_{s}$ is
a $\mathbb{P}^{5}$-fiberation over $\mathbf{K}$ outside of the subspace
$\mathbf{D}_{5}$. Here,
$\mathbf{D}_{5}\subset\mathbf{K}\setminus\mathbf{D}_{3}=\mathbf{H}[3]\setminus\widetilde{\mathbf{D}}_{3}$
parametrizes what are called _non-curvilinear_ points in $\mathbb{P}^{2}$
(Definition 4.1). Over $\mathbf{D}_{5}$, the fiber of $p_{s}$ parameterizes
stable sheaves whose support are rational quartic curves with a
$D_{4}$-singularity (Figure 1) and thus $p_{s}|_{p_{s}^{-1}(\mathbf{D}_{5})}$
is a $\mathbb{P}^{8}$-fiberation over $\mathbf{D}_{5}$ ((1) of Proposition
2.8). The aim of this paper is to resolve this singular locus by the
flattening of subschemes.

######  Theorem 1.1 (Theorem 4.7).

Let $\mathbf{R}_{4}$ be the closure of the space of rational quartic curves in
the moduli space $\mathbf{M}_{4}(\mathbb{P}^{2})$ of stable sheaves in
$\mathbb{P}^{2}$ with Hilbert polynomial $4m+1$. There exists an
$\mathrm{SL}(3,\mathbb{C})$-equivariant smooth resolution
$\widetilde{\mathbf{R}}_{4}$ of $\mathbf{R}_{4}$.

In fact, $\widetilde{\mathbf{R}}_{4}$ is a projective bundle over the blown-up
space $\widetilde{\mathbf{K}}$ of $\mathbf{K}$ along $\mathbf{D}_{5}$. As a
corollary, we set up the recipe for the computation of Chow ring of the space
$\widetilde{\mathbf{R}}_{4}$ (Proposition 4.8) and confirm the flip picture
(Corollary 4.9) suggested by Maruyama ([Mar73]).

The main idea of the proof of Theorem 1.1 is as follows. Motivated from the
canonical form of the planar rational quartic curve, we consider a rank $3$
vector bundle $\mathcal{F}$ on $\mathbf{K}$ whose fiber is generated by degree
$2$ parts of the defining equation of three points on $\mathbb{P}^{2}$. By
taking the projectivization of the second symmetric power
$\mathrm{Sym}^{2}\mathcal{F}=:\mathcal{V}$, we have a birational map

$\Phi:\mathbb{P}(\mathcal{V})\dashrightarrow\mathbb{P}(\mathcal{U})$

(induced by the inclusion map $\mathcal{V}\subset\mathcal{U}$) over
$\mathbf{K}$ where the generic fiber of the bundle $\mathcal{U}$ is generated
by the quartic forms passing through three points in $\mathbb{P}^{2}$. We show
that the non-torsion free locus of the quotient sheaf
$\mathcal{U}/\mathcal{V}$ is exactly the locus
$\mathbf{D}_{5}(\cong\mathbb{P}^{2})$ of the non-curvilinear points. Now by
considering the family of square ideals of three points in $\mathbb{P}^{2}$,
we see that $\mathbf{D}_{5}$ parameterizes the locus of non-flat family of
subschemes in $\mathbb{P}^{2}$. After the blow-up of $\mathbf{K}$ along
$\mathbf{D}_{5}$ and flattening of subschemes, we extend the map $\Phi$ as a
birational morphism. All of the aforementioned statements are in the
$\mathrm{SL}(3,\mathbb{C})=:\mathrm{SL}_{3}$-equivariant setting.
Additionally, it is remarkable that the Kodaira-Spencer map in the proof
Proposition 4.5 is an equivariant one and thus we easily detect the flat
limits parametrized by the blown-up space $\widetilde{\mathbf{K}}$.

###  1.3. Organization of the paper

In Section 2, we collect the well-known facts about the Hilbert scheme of
points, Bridgeland’s stability, and the symmetric power of coherent sheaves.
In Section 3, as a warm-up, we study the compactification of the space of
rational cubic curves in $\mathbb{P}^{2}$. Lastly, in Section 4, we provide a
proof of Theorem 1.1.

### Notations and conventions

  * •

Let $W_{n+1}$ be an $(n+1)$-dimensional complex vector space. We shortly write
$\mathbb{P}^{n}=\mathbb{P}W_{n+1}$.

  * •

The _projective bundle_ of a locally free sheaf $\mathcal{F}$ over $X$ is
defined by

$\mathbb{P}(\mathcal{F}):=\mathrm{Proj}\;(\mathrm{Sym}^{\bullet}(\mathcal{F}^{*}))\rightarrow
X,$

Hence $\mathbb{P}(\mathcal{F})$ is the space of one-dimensional subspaces of
$\mathcal{F}$.

  * •

$\mathbf{M}_{4}:=\mathbf{M}_{d}(\mathbb{P}^{2})$: The moduli space of stable
sheaves $F$ with Hilbert polynomial $\chi(F(m))=4m+1$.

  * •

$\mathbf{K}:=\mathbf{K}(3;2,3)$: The moduli space of quiver representations of
$3$-Kronecker quiver with the dimension vector $(2,3)$ (Notation 2.1).

  * •

$\mathbf{Q}_{4}:=\mathbb{P}(\mathcal{U})$, where $\mathcal{U}$ is a rank $12$
vector bundle over $\mathbf{K}$ (Definition 2.4).

  * •

$\mathbf{H}[n]:=\mathrm{Hilb}^{n}(\mathbb{P}^{2})$: The Hilbert scheme of
zero-dimensional subschemes of length $n$ on $\mathbb{P}^{2}$.

  * •

$\Rightarrow$ (resp. $\xhookrightarrow{}{}$, $\twoheadrightarrow$) denotes a
bundle (resp. injective, surjective) morphism of varieties or locally free
sheaves.

  * •

We often omit the pullback symbol in the pullback of locally free sheaves (or
vector bundles) along a morphism.

### Acknowledgements

The topic in this paper was originally proposed by D. Chen when K. Chung was
graduating ([Chu11, Chapter 6]), which provided an opportunity to explore
various compactifications of the space of rational curves. K. Chung expresses
deep gratitude to D. Chen. After writing the paper, the authors noticed some
overlap between Section 4.2 and [LFV21, Section 6]. We apologize for any
redundancy with [LFV21], although our results are based on the same
perspective as those in [Chu21].

##  2\. Preliminary

In this section, we collect very well-known facts about the Hilbert scheme of
points, the flip of moduli of stable objects and symmetric power of coherent
sheaves.

###  2.1. The Kronecker and Hilbert scheme

The moduli space of Kronecker modules is given by a geometric invariant
theoretic (GIT) quotient as follows. Let $\mathbf{K}(3;2,3)$ be the moduli
space of quiver representations of $3$-Kronecker quiver

$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$

with dimension vector $(2,3)$. It can be identified with the space of
isomorphism classes of stable sheaf homomorphisms

$\mathcal{O}_{\mathbb{P}^{2}}(-2)^{\oplus
2}\longrightarrow\mathcal{O}_{\mathbb{P}^{2}}(-1)^{\oplus 3}$

up to the action of the automorphism group
$G:=\mathrm{GL}_{d-2}\times\mathrm{GL}_{d-1}/\mathbb{C}^{*}$. Thus for two
vector spaces $E$ and $F$ of dimension $2$ and $3$ respectively and
$W_{3}^{*}=\mathrm{H}^{0}(\mathcal{O}_{\mathbb{P}^{2}}(1))$, the moduli space
of Kronecker modules is given by the GIT quotient ([Kin94])

$\mathbf{K}(3;2,3):=\mathrm{Hom}(F,W_{3}^{*}\otimes E)/\\!/G.$

######  Notation 2.1.

Let $\mathbf{K}:=\mathbf{K}(3;2,3)$.

The space $\mathbf{K}$ carries two bundles $\mathcal{E}$ and $\mathcal{F}$
with $\mathrm{rank}\,\mathcal{E}=2$ and $\mathrm{rank}\,\mathcal{F}=3$ such
that $\wedge^{2}\mathcal{E}\cong\wedge^{3}\mathcal{F}$. Furthermore, the
bundle $\mathcal{F}$ provides an embedding map
$i:\mathbf{K}\subset\mathrm{Gr}(3,\mathrm{Sym}^{2}W_{3}^{*})$ via the
inclusion map

$\mathcal{F}\subset\mathrm{Sym}^{2}W_{3}^{*}\otimes\mathcal{O}_{\mathbf{K}}.$
(1)

On the other hand, there exists a _universal complex_

$\mathbb{K}^{\bullet}:\mathcal{E}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-1)\stackrel{{\scriptstyle\phi}}{{\hookrightarrow}}\mathcal{F}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\mathcal{O}_{\mathbf{K}}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(2)$
(2)

over $\mathbf{K}\times\mathbb{P}^{2}$ such that $\phi$ is injective. The first
map $\phi$ in the complex $\mathbb{K}^{\bullet}$ in (2) provides the universal
resolution of the ideal of the Hilbert scheme $\mathbf{H}[3]$ of three points
on $\mathbb{P}^{2}$ excepting the locus $\widetilde{\mathbf{D}_{3}}$ of
collinear three points. Note that $\widetilde{\mathbf{D}_{3}}$ is isomorphic
to a $\mathbb{P}^{3}$-bundle over the space $\mathrm{Gr}(1,W_{3}^{*})$ of
lines, where each fiber $\mathbb{P}^{3}$ parameterizes triple points on a
line. In fact, the image of the map

$\pi:\mathbf{H}[3]\rightarrow\mathrm{Gr}(3,\mathrm{Sym}^{2}W_{3}^{*}),\;\;[Z]\mapsto\mathrm{H}^{0}(I_{Z}(2))$

is $\mathbf{K}$ and the map $\pi$ is a smooth blow-down with the exceptional
divisor $\widetilde{\mathbf{D}}_{3}$ ([LQZ03]). Also the restriction map of
$\pi$ along $\widetilde{\mathbf{D}}_{3}$ contracts the fiber $\mathbb{P}^{3}$.

######  Notation 2.2.

Let us denote the image of $\widetilde{\mathbf{D}}_{3}$ by
$\mathbf{D}_{3}(\cong\mathrm{Gr}(1,W_{3}^{*})$) in $\mathbf{K}$.

For each closed point $[Z]\in\mathbf{K}\setminus\mathbf{D}_{3}$, the fibers of
locally free sheaves $\mathcal{E}$ and $\mathcal{F}$ at $[Z]$ are given by

$\begin{split}\mathcal{F}_{[Z]}&=\mathrm{H}^{0}(\mathbb{P}^{2},I_{Z}(2))\subset\mathrm{H}^{0}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(2))=\mathrm{Sym}^{2}W_{3}^{*},\\\
\mathcal{E}_{[Z]}&=\text{ker}\left\\{\mathcal{F}_{[Z]}\otimes\mathrm{H}^{0}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1))\stackrel{{\scriptstyle
m}}{{\longrightarrow}}\mathrm{H}^{0}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(3))\right\\},\end{split}$

where $m$ is the restriction of the canonical multiplication map
$m:\mathrm{Sym}^{2}W_{3}^{*}\otimes
W_{3}^{*}\rightarrow\mathrm{Sym}^{3}W_{3}^{*}$. But the complex
$\mathbb{K}^{\bullet}$ in (2) is not acyclic (i.e., $h^{0}\neq 0$) over
$\mathbf{D}_{3}$. For each closed point $[l]\in\mathbf{D}_{3}$, let
$\text{coker}(\phi)_{[l]}:=E_{[l]}$ be the cokernel of $\phi_{[l]}$. Then
$E_{[l]}$ fits into the short exact sequence:

$0\rightarrow{\mathcal{O}_{l}(-1)}\rightarrow{E_{[l]}}\rightarrow{\mathcal{O}_{\mathbb{P}^{2}}(1)}\rightarrow
0.$ (3)

Furthermore, the map $\psi_{[l]}$ in the complex $\mathbb{K}^{\bullet}$
decomposes into

${{\mathcal{F}_{[l]}\otimes\mathcal{O}_{\mathbb{P}^{2}}}}$${\mathcal{O}_{\mathbb{P}^{2}}(2)}$${{E_{[l]}}}$${\mathcal{O}_{\mathbb{P}^{2}}(1).}$$\scriptstyle{\psi_{[l]}}$
(4)

######  Remark 2.3.

The complex in (2) is a special case of the following one. Let
$\phi:\mathcal{E}_{0}\rightarrow\mathcal{E}_{1}$ be a morphism of locally free
sheaves $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ on a smooth projective variety
$X$ with $\mathrm{rank}\,\mathcal{E}_{0}+1=\mathrm{rank}\,\mathcal{E}_{1}=n$.
There exists an _Eagon-Northcott complex_ associated to the dual map
$\phi^{t}:\mathcal{E}_{1}^{*}\rightarrow\mathcal{E}_{0}^{*}$ ([EN62]). By
using an isomorphism
$\wedge^{n-1}\mathcal{E}_{1}\cong\mathcal{E}_{1}^{*}\otimes\wedge^{n}\mathcal{E}_{1}$,
one can have a complex

$\mathcal{E}_{0}\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\mathcal{E}_{1}\stackrel{{\scriptstyle\wedge^{n-1}\phi^{t}}}{{\longrightarrow}}(\wedge^{n}\mathcal{E}_{1})\otimes\wedge^{n-1}\mathcal{E}_{0}^{*}.$

###  2.2. Moduli space of stable objects and its flip

The moduli space of semistable objects on $\mathbb{P}^{2}$ has been
extensively studied in the context of birational geometry ([BMW14]). In this
paper, we briefly discuss the chamber structure of Bridgeland’s stability
space and its wall-crossings.

We define the potential function for $s\in\mathbb{R}$ and $t>0$ as follows:

$Z_{s,t}(E)=-\int_{\mathbb{P}^{2}}e^{-(s+it)H}\cdot
ch(E)=-\left(ch_{2}-sc_{1}+\frac{1}{2}r(s^{2}-t^{2})\right)+it(c_{1}-rs),$

where $(r,c_{1},ch_{2})=(r(E),c_{1}(E),ch_{2}(E))$ and
$H=c_{1}(\mathcal{O}_{\mathbb{P}^{2}}(1))$. We can determine the stability of
objects in the heart $\mathcal{A}_{s}$ of a certain $t$-structure (depending
only on $s$) in $\mathrm{D}^{b}(\mathbb{P}^{2})$ using the slope function:

$\mu_{s,t}:\mathrm{K}(\mathbb{P}^{2})\to\mathbb{R},\;\mu_{s,t}(E)=-\frac{\mathrm{Re}(Z_{s,t}(E))}{\mathrm{Im}(Z_{s,t}(E))}=\frac{ch_{2}-sc_{1}+\frac{1}{2}r(s^{2}-t^{2})}{t(c_{1}-rs)}.$

Also the stability space (identified with the upper half plane
$\mathbb{H}=\\{(s,t)\in\mathbb{R}^{2}\;|\;t>0\\}$) can be decomposed into a
finite number of chambers. Let $\mathrm{M}_{\mu_{s,t}}(v)$ be the moduli space
of Bridgeland stable objects in $\mathcal{A}_{s}$ with a fixed type
$v=(r,c_{1},ch_{2})\in\mathrm{A}^{*}(\mathbb{P}^{2})$. The space
$\mathrm{M}_{\mu_{s,t}}(v)$ varies only if $(s,t)$ moves from one chamber to
another.

In our case, let us fix $v(F)=(0,4,-5)\in\mathrm{A}^{*}(\mathbb{P}^{2})$ (and
thus $\chi(F(m))=4m+1$).

######  Definition 2.4.

Let
$\mathcal{U}:=p_{*}(\text{coker}(\phi)\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(2))$
in (2) be the locally free sheaf on $\mathbf{K}$, where
$p:\mathbf{Q}_{4}:=\mathbb{P}(\mathcal{U})\rightarrow\mathbf{K}$ is the
canonical bundle morphism.

######  Remark 2.5.

The projectivized bundle $\mathbb{P}(\mathcal{U})$, when restricted to the
open subset $\mathbf{K}\setminus\mathbf{D}_{3}$, is isomorphic to the relative
Hilbert scheme parameterizing three non-collinear points on plane quartic
curves, as shown in [He98, Section 4.4].

######  Proposition 2.6 ([Woo13, Lemma 6.9]).

Under the above notations and definitions,

  1. (1)

The walls dividing chambers in $\mathbb{H}$ are semi-circles with the common
center $(-\frac{5}{4},0)$.

  2. (2)

There is only one numerical wall when the radius of the semicircle is
$R=\frac{7}{4}$.

  3. (3)

There are two different moduli spaces of stable objects:

$\mathrm{M}_{\mu_{s,t}}(v)\cong\begin{cases}\mathbf{Q}_{4},\;t\ll 1,\\\
\mathbf{M}_{4},\;t\gg 1.\end{cases}$

The universal subbundle homomorphism
$\mathcal{O}_{\mathbf{Q}_{4}}(-1)\hookrightarrow p^{*}\mathcal{U}$ determines
a universal family of stable complexes

$\mathbb{F}_{4}:=[\mathcal{O}_{\mathbf{Q}_{4}\times\mathbb{P}^{2}}\rightarrow\pi_{1}^{*}\mathcal{O}_{\mathbf{Q}_{4}}(1)\otimes(p\times\text{id})^{*}(\text{coker}(\phi)\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(2))].$
(5)

As modifying of the complex $\mathbb{F}$ in (5), there exists a flip diagram
between $\mathbf{Q}_{4}$ and $\mathbf{M}_{4}$ such that the flipping loci are
given by

${\mathbf{Q}_{4}}$${{}^{\text{flip}}}$${\mathbf{M}_{4}}$${F^{-}=\text{Fl}(W_{3})}$${F^{+}=\mathcal{C}_{4}}$${\mathbb{P}^{2}}$

  * •

$\mathbf{F}^{-}$ is isomorphic to the _universal lines_
$\text{Fl}(W_{3})=\\{(q,l)\mid q\in l\in\mathrm{Gr}(1,W_{3}^{*})\\}$ and

  * •

$\mathbf{F}^{+}$ is isomorphic to the _universal quartic curves_
$\mathcal{C}_{4}=\\{(q,C)\mid q\in C\in|\mathcal{O}_{\mathbb{P}^{2}}(4)|\\}$.

We remark that the flipping locus $\mathbf{F}^{+}$ parameterizes stable
sheaves $F$ satisfying $h^{0}(F)=2$. The complex
$F:=[\mathcal{O}_{\mathbb{P}^{2}}\stackrel{{\scriptstyle
s}}{{\rightarrow}}I_{Z|\mathbb{P}^{2}}(4)]\in\mathbf{Q}_{4}\setminus\mathbf{F}^{-}$
bijectively associates its derived dual
$F^{\vee}\in\mathbf{M}_{4}\setminus\mathbf{F}^{+}$.

######  Proposition 2.7.

Let $[l]\in\mathbf{D}_{3}=\mathrm{Gr}(1,W_{3}^{*})$ and

$s=(s_{1},s_{2})\in\mathrm{H}^{0}(E_{[l]}(2))=\mathrm{H}^{0}(\mathcal{O}_{l}(1))\oplus\mathrm{H}^{0}(\mathcal{O}_{\mathbb{P}^{2}}(3))$

such that $E_{[l]}$ in (3). Then,

  1. (1)

$\mathbf{F}^{-}$ is contained in $p^{-1}(\mathbf{D}_{3})$.

  2. (2)

$s_{2}=0$ if and only if
$[\mathbb{F}_{4}|_{([l],s)\times\mathbb{P}^{2}}]\in\mathbf{F}^{-}$, where the
stable complex fits into the exact triangle

$0\rightarrow[\mathcal{O}_{\mathbb{P}^{2}}\stackrel{{\scriptstyle
s_{1}}}{{\longrightarrow}}\mathcal{O}_{l}(1)]\rightarrow\mathbb{F}_{4}|_{([l],s)\times\mathbb{P}^{2}}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(3)[1]\rightarrow
0.$

  3. (3)

If $s_{2}\neq 0$, then the derived dual
$\mathbb{F}_{4}|_{([l],s)\times\mathbb{P}^{2}}^{\vee}$ is quasi-isomorphic to
a stable sheave $F$ fitting into the exact sequence

$0\rightarrow{\mathcal{O}_{C}}\rightarrow{F}\rightarrow{\mathcal{O}_{l}}\rightarrow
0$ (6)

for a cubic curve $C=\text{Supp}(\text{coker}(s_{2}))$. Thus $h^{0}(F)=1$.

###### Proof.

(1) and (2): [CM17, Lemma 3.9 and Lemma 3.7].

(3) When the diagram (9) in [CM17] is read in reverse, one can obtain the
exact sequence (6). ∎

Recall that $\mathbf{R}_{4}$ is the closure of the space of rational quartic
curves in $\mathbf{M}_{4}$. From [Chu21, Proposition 3.3], $h^{0}(F)=1$ for
each point $[F]\in\mathbf{R}_{4}$ and thus there is an embedding
$\mathbf{R}_{4}\subset\mathbf{M}_{4}\setminus\mathbf{F}^{-}$.

######  Proposition 2.8 ([Chu21, Proposition 3.6 and Proposition 4.1]).

Let $\pi$ be the composition of the maps

$\pi:\mathbf{R}_{4}\subset\mathbf{M}_{4}\setminus\mathbf{F}^{-}\cong\mathbf{Q}_{4}\setminus\mathbf{F}^{+}\rightarrow\mathbf{K}.$

Then the map $\pi$ is a generic $\mathbb{P}^{5}$-fibration over $\mathbf{K}$
and the special fiber is described as follows.

  1. (1)

$\pi$ is a $\mathbb{P}^{8}$-fibration map over the locus $\mathbf{D}_{5}$ of
non-curvilinear points (For the explicit definition, see Definition 4.1) such
that the fiber $\mathbb{P}^{8}$ parameterizes the plane quartic curves
singular at a point parameterized by $\mathbf{D}_{5}$.

  2. (2)

For any $[l]\in\mathbf{D}_{3}$, the fiber $\pi^{-1}([l])\cong\mathbb{P}^{5}$
parametrizes a stable sheaf $F$ that fits into a _unique_ exact sequence of
the form $0\rightarrow{\mathcal{O}_{C\cdot
l}}\rightarrow{F}\rightarrow{\mathcal{O}_{l}}\rightarrow 0$, where $C$ is a
conic parameterized by the fiber $\mathbb{P}^{5}$ (cf. Proposition 2.7).

The rational curves in (1) of Proposition 2.8 generically have
$D_{4}$-singularities (Figure 1). For discussions on singularities of quartic
plane curves, we may refer to [Hui79] and [CKPU13, Chapter 9]. The main goal
of this paper is to resolve this locus by means of elementary modification of
sheaves (bundles).

Figure 1. A rational quartic plane curve with a $D_{4}$-singularity

###  2.3. Symmetric powers and its basics of $\mathrm{SL}_{3}$-representation

Let $\mathcal{F}^{\otimes k}$ be the tensor power of a coherent sheaf
$\mathcal{F}$ on a smooth projective variety $X$. The $k$-th symmetric power
of $\mathcal{F}$, denoted by $\operatorname{Sym}^{k}\mathcal{F}$, is defined
as the quotient of the tensor power $\mathcal{F}^{\otimes k}$ by the ideal
sheaf generated by all elements of the form $s_{1}\otimes\cdots\otimes
s_{k}-\sigma(s_{1})\otimes\cdots\otimes\sigma(s_{k})$, where
$s_{1},\dots,s_{k}$ are local sections of $\mathcal{F}$ and $\sigma$ is any
element of the symmetric group $S_{n}$ of $n$-letters. Let us denote the
equivalence class by $s_{1}\cdot s_{2}\cdots\cdot s_{k}:=[s_{1}\otimes
s_{2}\otimes\cdots\otimes s_{k}]\in\operatorname{Sym}^{k}\mathcal{F}$.

######  Lemma 2.9.

Let $\mathcal{E}$, $\mathcal{F}$, and $\mathcal{H}$ be coherent sheaves on
$X$. Consider an exact sequence
${\mathcal{E}}\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}{\mathcal{F}}\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}{\mathcal{H}}\longrightarrow
0$. Then, for all $k\geq 1$, we obtain an exact sequence

$\mathcal{E}\otimes\mathrm{Sym}^{k-1}\mathcal{F}\stackrel{{\scriptstyle
s(\delta)}}{{\longrightarrow}}\mathrm{Sym}^{k}\mathcal{F}\stackrel{{\scriptstyle\text{sym}^{k}(\epsilon)}}{{\longrightarrow}}\mathrm{Sym}^{k}\mathcal{H}\rightarrow
0$

where $s(\delta)$ is defined by

$e\otimes(f_{1}\cdot f_{2}\cdots f_{k-1})\mapsto\sum_{i=1}^{k}(f_{1}\cdot
f_{2}\cdots\delta(e)\cdots f_{k-1}).$

For our purpose, we need a standard $\mathrm{SL}_{3}$-representation theory
related with symmetric power spaces. We summarize the contents of Section 13.3
in [FH91]. Let $V$ be a vector space with dimension $\dim V=3$. Then the
standard representation of $\mathrm{Sym}^{2}(\mathrm{Sym}^{2}V)$ is isomorphic
to the space

$\begin{split}\mathrm{Sym}^{2}(\mathrm{Sym}^{2}V)&\cong\mathrm{Sym}^{4}V\oplus\mathrm{Sym}^{2}V^{*}\\\
&\cong\mathrm{Sym}^{4}V\oplus\mathrm{Sym}^{2}(\wedge^{2}V)\end{split}$

where the kernel of the multiplication map
$m:\mathrm{Sym}^{2}(\mathrm{Sym}^{2}V)\twoheadrightarrow\mathrm{Sym}^{4}V$ is
identified with
$\text{ker}(m)=\mathrm{Sym}^{2}(\wedge^{2}V)\cong\mathrm{Sym}^{2}V^{*}$ under
a fixed isomorphism $\wedge^{2}V\cong V^{*}\otimes\wedge^{3}V$. Here the
embedding is given by

$\mathrm{Sym}^{2}(\wedge^{2}V)\subset\mathrm{Sym}^{2}(\mathrm{Sym}^{2}V),\;(u\wedge
v)\cdot(w\wedge z)\mapsto(u\cdot w)\cdot(v\cdot z)-(u\cdot z)\cdot(v\cdot w).$

######  Example 2.10.

Let $(I_{Z})_{2}=\langle x^{2},xy,y^{2}\rangle$ be the degree $2$ part of the
ideal of the non-curvilinear point $Z$ supported at
$\text{Supp}(Z)=\\{[0:0:1]\\}$. As a $\mathbb{C}$-vector space, $(I_{Z})_{2}$
is a subvector space of $\mathrm{Sym}^{2}V_{3}$ for
$V_{3}=\text{span}_{\mathbb{C}}\\{x,y,z\\}$. One may choose a basis of
$\mathrm{Sym}^{2}(\langle x^{2},xy,y^{2}\rangle)$ as

$\left\\{x^{2}\cdot x^{2},\;x^{2}\cdot(xy),\;x^{2}\cdot
y^{2},\;(xy)\cdot(xy),\;(xy)\cdot y^{2},\;y^{2}\cdot y^{2}\right\\},$

which is a subvector space of $\mathrm{Sym}^{2}(\mathrm{Sym}^{2}V_{3})$. Note
that $x^{2}\cdot
y^{2}\neq(xy)\cdot(xy)\in\mathrm{Sym}^{2}(\mathrm{Sym}^{2}V_{3})$. However,
under the canonical multiplication map
$m:\mathrm{Sym}^{2}(\mathrm{Sym}^{2}V_{3})\twoheadrightarrow\mathrm{Sym}^{4}V_{3}$,
they have the same image, i.e., $m(x^{2}\cdot y^{2})=m((xy)\cdot(xy))$.
Therefore, we have

$(x\wedge y)\cdot(x\wedge y)=x^{2}\cdot y^{2}-(xy)\cdot(xy)\in\ker(m).$

##  3\. Warm-up: Degree $3$ case

In this section, we describe the space $\mathbf{R}_{3}$ as a projective bundle
based on the results of [CC11]. Even though the result is very elementary, the
purpose of this writing is to provide motivation for the degree $4$ case,
which will be discussed in Section 4.

###  3.1. Rational cubic curves space and its universal complex

Consider a cubic curve $C$ in $\mathbb{P}^{2}=\mathbb{P}(W_{3})$ defined by
the cubic equation $f(x,y,z)=0$. A cubic curve $C$ is rational if it has a
singular point by the degree-genus formula. The cubic curve $C$ passes through
the point $q=[0:0:1]$ if and only if $f(x,y,z)$ can be written in the form

$f(x,y,z)=g_{3}(x,y)+zg_{2}(x,y)+z^{2}g_{1}(x,y),$ (7)

where $g_{i}(x,y)$ is a homogeneous polynomial of $x,y$ and $\deg g(x,y)=i$.
Furthermore, $C$ is singular at the point $q$ if and only if $g_{1}(x,y)=0$ in
(7). We describe how one can construct the parameter space of rational cubic
curves through the canonical form (7). Let $\mathcal{E}_{i}$ be the locally
free sheaf whose fiber corresponds to cubic forms vanishing of order $i$ at a
point in $\mathbb{P}^{2}$. Let $\mathbf{D}=\mathrm{Gr}(2,W_{3}^{*})$ be the
Grassmannian of points in $\mathbb{P}^{2}$. By taking symmetric powers of the
universal bundle sequence
$0\rightarrow{\mathcal{D}}\rightarrow{W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}}}\rightarrow{\mathcal{Q}}\rightarrow
0$, we obtain a commutative diagram:

$\textstyle{W_{3}^{*}\otimes\mathcal{Q}^{\otimes
2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{3}^{*}\otimes\mathcal{Q}^{\otimes
2}}$$\textstyle{\mathrm{Sym}^{3}\mathcal{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sym}^{3}W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sym}^{2}W_{3}^{*}\otimes\mathcal{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sym}^{3}\mathcal{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sym}^{2}\mathcal{D}\otimes\mathcal{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(8)

such that the sheaf $\mathcal{E}_{2}$ fits into the bottom row of the diagram
(8).

######  Proposition 3.1.

Under the above notations and definitions,

  1. (1)

There exist inclusions of locally free sheaves on $\mathbf{D}$:

$\mathcal{E}_{2}\subset\mathcal{E}_{1}\subset\mathrm{Sym}^{3}W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}}.$

Furthermore, the moduli space $\mathbf{M}_{3}$ (resp. $\mathbf{R}_{3}$) is
isomorphic to

$\mathbf{M}_{3}\cong\mathbb{P}(\mathcal{E}_{1}),\;(\text{resp}.\;\mathbf{R}_{3}\cong\mathbb{P}(\mathcal{E}_{2}))$

and the quotient $\mathcal{E}_{1}/\mathcal{E}_{2}$ is isomorphic to a locally
free sheaf
$\mathcal{E}_{1}/\mathcal{E}_{2}\cong\mathcal{D}\otimes\mathcal{O}_{\mathbf{D}}(2)$.

  2. (2)

The universal family of stable complex over $\mathbf{R}_{3}$ is given by the
derived dual

$\mathbb{F}_{3}:=[\mathcal{O}_{\mathbf{R}_{3}\times\mathbb{P}^{2}}\rightarrow\pi_{1}^{*}\mathcal{O}_{\mathbf{R}_{3}}(1)\otimes(p\times\text{id})^{*}(\mathcal{I}_{\mathcal{W}}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(3))]^{\vee}$
(9)

where $\mathcal{W}$ is the universal subscheme of a point over $\mathbf{D}$
and $p:\mathbb{P}(\mathcal{E}_{2})\rightarrow\mathbf{D}$ is the canonical
projection map.

###### Proof.

(1) One can easily see that the kernel of the composition map of the middle-
upper map of the diagram (8) and the canonical surjective map
$W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}}\twoheadrightarrow\mathcal{Q}$
twisted by $\mathcal{Q}^{\otimes 2}$ is the locally free sheaf

$\mathcal{E}_{1}=\text{Ker}\left\\{\mathrm{Sym}^{3}W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}}\twoheadrightarrow
W_{3}^{*}\otimes\mathcal{Q}^{\otimes 2}\twoheadrightarrow\mathcal{Q}^{\otimes
3}\right\\}.$

Thus we prove the inclusion relation. Also the isomorphisms come from the
first paragraph of [CC11]. The quotient sheaf
$\mathcal{E}_{1}/\mathcal{E}_{2}$ is finally obtained by analyzing the
canonical form in (7).

(3) One can construct the universal complex over $\mathbf{M}_{3}$ by
relativizing the following nested exact sequence

$0\rightarrow{\mathcal{O}_{\mathbb{P}^{2}}(-3)=I_{C|\mathbb{P}^{2}}}\rightarrow{I_{q|\mathbb{P}^{2}}}\rightarrow{I_{q|C}}\rightarrow
0$

of ideals for the pair $q\in C$ for some point $q$ and cubic curve $C$ in
$\mathbb{P}^{2}$. The claimed universal complex in (9) is nothing but the
pull-back of this complex along the inclusion map
$\mathbb{P}(\mathcal{E}_{2})\times\mathbb{P}^{2}\subset\mathbb{P}(\mathcal{E}_{1})\times\mathbb{P}^{2}$.
∎

##  4\. Degree $4$ case

In this section, we first construct a birational model of $\mathbf{R}_{4}$,
which is a projectivization of a symmetrized bundle on the Kronecker modules
space (Proposition 4.4). Secondly, by studying the undefined loci and
flattening of a family of subschemes (Proposition 4.5), we modify the bundle
and thus obtain an extended birational morphism into $\mathbf{R}_{4}$ (Theorem
4.7). Finally, we explain the contraction of the exceptional locus into
$\mathbf{R}_{4}$. All of these works are an $\mathrm{SL}_{3}$-invariant one.

###  4.1. Universal bundles, symmetrization and its projectivization

In this subsection, we construct a birational map between a
$\mathbb{P}^{5}$-projective bundle over $\mathbf{K}$ and the rational curves
space $\mathbf{R}_{4}$ (Proposition 4.4). The canonical form ([Chu21, Section
3.2]) of the rational quartic plane curve suggests that a natural candidate
for such a bundle is the symmetric product $\mathrm{Sym}^{2}\mathcal{F}$ of
the universal subbundle $\mathcal{F}$ in (1). Let

$\mathrm{Sym}^{2}\mathcal{F}\subset\mathcal{F}\otimes\mathcal{F}$ (10)

be the inclusion map induced by $v\cdot w\mapsto\frac{1}{2}(v\otimes
w+w\otimes v)$ for $v,w\in\mathcal{F}$. As composing the inclusion map (10)
and the second map in $\psi$ in (2) tensored with $\mathcal{F}\boxtimes(-)$,
we have a morphism over $\mathbf{K}\times\mathbb{P}^{2}$:

$k:\mathrm{Sym}^{2}\mathcal{F}\subset\mathcal{F}\otimes\mathcal{F}\stackrel{{\scriptstyle\text{id}\otimes\psi}}{{\longrightarrow}}\mathcal{F}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(2).$
(11)

Now by the tensoring the line bundle $\mathcal{O}_{\mathbb{P}^{2}}(2)$ in the
universal complex (2) and combining with the map $k$ in equation (11), we get
a commutative diagram:

$\textstyle{\mathrm{Sym}^{2}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\scriptstyle{(\psi\otimes\text{id})\circ
k}$$\textstyle{\mathcal{F}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi\otimes\text{id}}$$\scriptstyle{s}$$\textstyle{\mathcal{O}_{\mathbf{K}}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(4)}$$\textstyle{\mathcal{M}:=\text{coker}(\phi)\otimes\mathcal{O}_{\mathbb{P}^{2}}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\psi}\otimes\text{id}}$
(12)

over $\mathbf{K}\times\mathbb{P}^{2}$. The push-forward along the projection
map $p:\mathbf{K}\times\mathbb{P}^{2}\rightarrow\mathbf{K}$ of the diagram
(12) provides

$\textstyle{p_{*}(\mathrm{Sym}^{2}p^{*}\mathcal{F})=\mathrm{Sym}^{2}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{s}}$$\textstyle{p_{*}\mathcal{O}_{\mathbb{P}^{2}}(4)=\mathrm{Sym}^{4}W_{3}^{*}\otimes\mathcal{O}_{\mathbf{K}}}$$\textstyle{p_{*}\mathcal{M}=\mathcal{U}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(13)

We need to introduce a special locus in $\mathbf{K}$ for the analysis of the
induced morphism $\overline{s}$ in (13), which will turn out to be a
degeneracy locus.

######  Definition 4.1.

Let us define $\mathbf{D}_{5}$ as the locus of _non-curvilinear_ points in
$\mathbb{P}^{2}$ (with the reduced scheme structure) that are
$\mathrm{SL}_{3}$-orbits of the ideal $I_{q}^{2}=\langle
x^{2},xy,y^{2}\rangle$ for the coordinate point $q=[0:0:1]$.

In an extrinsic sense, one can expect that $\mathbf{D}_{5}$ is isomorphic to
$\mathrm{Gr}(2,W_{3}^{*})$.

######  Lemma 4.2.

Let $\mathbf{D}:=\mathrm{Gr}(2,W_{3}^{*})$ be the Grassmannian of points in
the projective plane $\mathbb{P}^{2}=\mathbb{P}(W_{3})$. Then there exists an
embedding $j:\mathbf{D}\hookrightarrow\mathbf{K}$ whose image is
$\mathbf{D}_{5}$.

###### Proof.

Taking the second symmetric power of the universal sequence
$0\rightarrow{\mathcal{D}}\rightarrow{W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}}}\rightarrow{\mathcal{Q}}\rightarrow
0$ of the Grassmannian $\mathbf{D}$, we obtain the following sequence:

$0\longrightarrow\mathrm{Sym}^{2}\mathcal{D}\longrightarrow\mathrm{Sym}^{2}W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}}\longrightarrow(W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}})\otimes\mathcal{Q}\longrightarrow
0,$ (14)

which defines an embedding map into
$\mathrm{Gr}(3,\mathrm{Sym}^{2}W_{3}^{*})$. As checking the image fiberwisely,
the embedding map factors through $\mathbf{D}_{5}\subset\mathbf{K}$. ∎

######  Remark 4.3.

When we identify $\mathbf{K}$ with the Hilbert scheme $\mathbf{H}[3]$ of three
points in $\mathbb{P}^{2}=\mathbb{P}^{2}(W_{3})$, one can obtain a family of
subschemes of _non-curvilinear_ points parametrized by $\mathbf{D}$. Let
$\mathcal{W}$ be the universal family of points in $\mathbb{P}^{2}$ over
$\mathbf{D}$. Then the relative Koszul complex of the ideal sheaf
$\mathcal{I}_{\mathcal{W}}$ is acyclic:

$0\rightarrow\wedge^{2}\mathcal{D}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-2)\rightarrow\mathcal{D}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-1)\rightarrow\mathcal{I}_{\mathcal{W}}\rightarrow
0.$ (15)

By taking the second symmetric power of the sequence (15), we have an exact
sequence

$0\rightarrow(\wedge^{2}\mathcal{D}\otimes\mathcal{D})\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-3)\rightarrow\mathrm{Sym}^{2}\mathcal{D}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-2)\rightarrow\mathrm{Sym}^{2}\mathcal{I}_{\mathcal{W}}=\mathcal{I}^{2}_{\mathcal{W}}\rightarrow
0.$

Since
$\mathcal{I}^{2}_{\mathcal{W}}\subset\mathcal{I}_{\mathcal{W}}\subset\mathcal{O}_{\mathbf{D}\times\mathbb{P}^{2}}$,
one can easily check that $\mathcal{I}^{2}_{\mathcal{W}}$ is the flat family
of non-curvilinear points.

Now we are ready to state and prove one of the main propositions. Recall that
$\mathbf{R}_{4}$ is the closure of the space of rational quartic curves in
$\mathbb{P}^{2}$ under $\mathbf{M}_{4}$. Also
$\mathbf{R}_{4}\subset\mathbf{Q}_{4}\setminus\mathbf{F}^{-}=\mathbb{P}(\mathcal{U})\setminus\mathbf{F}^{-}$
where $\mathbf{F}^{-}$ is the flipping locus in $\mathbf{Q}_{4}$ (Section
2.2).

######  Proposition 4.4.

Let $\mathcal{V}:=\mathrm{Sym}^{2}\mathcal{F}$ in equation (13) and
$\mathcal{V}^{\circ}$ be the restriction of $\mathcal{V}$ along the open
subset $\mathbf{K}\setminus\mathbf{D}_{5}$. Then there exists an embedding map

$\Psi:\mathbb{P}(\mathcal{V}^{\circ})\xhookrightarrow{}{}\mathbb{P}(\mathcal{U})\setminus\mathbf{F}^{-}$

such that the image of $\Psi$ is birational to the space $\mathbf{R}_{4}$.

###### Proof.

Note that $\mathbf{D}_{3}\cap\mathbf{D}_{5}=\emptyset$ by its definition. For
each point $[Z]\in\mathbf{K}\setminus(\mathbf{D}_{3}\sqcup\mathbf{D}_{5})$,
the map $\overline{s}$ in (13) is nothing but the canonical isomorphism map

$\mathcal{V}_{[Z]}=\mathrm{Sym}^{2}\mathrm{H}^{0}(I_{Z}(2))\cong\mathrm{H}^{0}(I_{Z}^{2}(4))\subset\mathrm{H}^{0}(I_{Z}(4))=\mathcal{U}_{[Z]},$

where the second map comes from the inclusion $I_{Z}^{2}\subset I_{Z}$. Since
we can easily check that $h^{0}(I_{Z}^{2}(4))=6$ for each representative $[Z]$
of the $\mathrm{SL}_{3}$-orbits, $\overline{s}$ is injective.

For the closed point $[l]\in\mathbf{D}_{3}(=\mathrm{Gr}(1,W_{3}^{*}))$, then
the fiber of $\mathcal{U}_{[l]}=\mathrm{H}^{0}(E_{[l]}(2))$ (cf. (3)) fits
into the exact sequence

$0\rightarrow{\mathrm{H}^{0}(\mathcal{O}_{l}(1))}\rightarrow{\mathcal{U}_{[l]}}\rightarrow{\mathrm{H}^{0}(\mathcal{O}_{\mathbb{P}^{2}}(3))}\rightarrow
0.$

Also the map $\mathcal{U}_{[l]}\rightarrow
p_{*}\mathcal{O}_{\mathbb{P}^{2}}(4)_{[l]}$ in (13) factors through the
multiplication map
$l\cdot(-):\mathrm{H}^{0}(\mathcal{O}_{\mathbb{P}^{2}}(3))\rightarrow\mathrm{H}^{0}(\mathcal{O}_{\mathbb{P}^{2}}(4))\cong\mathrm{Sym}^{4}W_{3}^{*}$
(cf. (4)). Up to an $\mathrm{SL}_{3}$-equivalence, let $l=\langle x\rangle$.
Also we assume that
$\mathrm{H}^{0}(\mathcal{F}_{[l]})=\text{span}_{\mathbb{C}}(x^{2},xy,xz)$.
Then the image of $\mathcal{V}_{[l]}$ by the map $\overline{s}_{[l]}$ in
$\mathrm{Sym}^{4}W_{3}^{*}$ is $x^{2}\cdot\mathrm{Sym}^{2}W_{3}^{*}$ and thus
it is injective. Additionally, the image has a nonzero cubic part and hence it
lies the complement $\mathbb{P}(\mathcal{U})\setminus\mathbf{F}^{-}$ ((2) of
Proposition 2.8).

Lastly, for the ideal $I_{q}=\langle
x,y\rangle\in\mathbf{D}(\cong\mathbf{D}_{5})$, the image by $\overline{s}$ is
given by
$\mathrm{H}^{0}(\mathbb{P}^{2},I_{q}^{4}(4))\cong\mathrm{Sym}^{4}\langle
x,y\rangle$ and thus it is not injective. Since $\mathbf{D}_{5}$ is transitive
under the $\mathrm{SL}_{3}$-action, this property holds for every point in
$\mathbf{D}_{5}$. ∎

From the proof of Proposition 4.4, the map $\overline{s}$ in (13) is
generically injective (hence injective as sheaf homomorphism). Furthermore,
the map $\overline{s}$ decomposes into the following way. By taking the
symmetric power of the equation (1) and projecting into the first component
(Section 2.3), we have

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.88617pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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24.88618pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
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84.20511pt\raise-81.6878pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
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By its construction, the map $\overline{s}$ in (13) is simply the composition
of the map $p_{1}$ followed by the inclusion map $i$. The proof of Proposition
4.4 tells us that the map $\overline{s}=p_{1}\circ i$ is injective outside
$\mathbf{D}_{5}$.

###  4.2. Modification of the symmetrized bundle via ideal flattening

In this subsection, by modifying the bundle
$\mathcal{V}=\mathrm{Sym}^{2}\mathcal{F}$ on $\mathbf{K}$ along
$\mathbf{D}_{5}$, we obtain a subbundle $\mathcal{V}^{\prime}$ of
$\mathcal{U}$ such that the quotient sheaf $\mathcal{U}/\mathcal{V}^{\prime}$
is locally free (Theorem 4.7). From this, we can deduce the existence of a
morphism $\mathbb{P}(\mathcal{V}^{\prime})\rightarrow\mathbb{P}(\mathcal{U})$
and therefore a morphism to the space $\mathbf{R}_{4}$.

Let $\mathrm{im}((\psi\otimes\text{id})\circ k)$ be the image of
$\mathrm{Sym}^{2}\mathcal{F}$ in (12). Since our modification will be done
along the complement of $\mathbf{D}_{3}=\mathrm{Gr}(1,W_{3}^{*})$ in
$\mathbf{K}$ (Section 2.1), we can assume that
$\mathrm{im}((\psi\otimes\text{id})\circ
k)\subset\mathcal{M}\cong\mathcal{I}_{\mathcal{Z}}(4)$ where $\mathcal{Z}$ is
the universal subscheme over $\mathbf{K}$ and thus the image becomes a twisted
ideal sheaf

$\mathrm{im}((\psi\otimes\text{id})\circ
k)\cong\mathcal{I}_{\mathcal{S}}(4)\subset\mathcal{I}_{\mathcal{Z}}(4)\subset\mathcal{O}_{\mathbf{K}}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(4)$

for a subscheme $\mathcal{S}\subset\mathbf{K}\times\mathbb{P}^{2}$. As shown
in the proof of Proposition 4.4, the fibers over $\mathbf{K}$ are determined
by

$\mathcal{I}_{\mathcal{S}}|_{[Z]\times\mathbb{P}^{2}}=\begin{cases}I_{Z}^{2},\;[Z]\in\mathbf{K}\setminus\mathbf{D}_{5},\\\
I_{q}^{4},\;[Z]=[q^{2}]\in\mathbf{D}_{5}.\end{cases}$ (16)

From now on, we will denote $q^{k}$ by the subscheme defined by the ideal
$I_{q}^{k}$ for a point $q\in\mathbb{P}^{2}$. Since the first (resp. second)
case in (16) has constant Hilbert polynomial $10$ (resp. $9$), $\mathcal{S}$
is a flat family of subschemes in $\mathbb{P}^{2}$ over
$\mathbf{K}\setminus\mathbf{D}_{5}$. We will blow up $\mathbf{K}$ along
$\mathbf{D}_{5}$ and extend the flat family as follows: Let
$\pi:\widetilde{\mathbf{K}}\rightarrow\mathbf{K}$ be the blow-up of
$\mathbf{K}$ along $\mathbf{D}_{5}$. Pull back $\mathcal{S}$ along the map
$\pi\times\text{id}:\widetilde{\mathbf{K}}\times\mathbb{P}^{2}\rightarrow\mathbf{K}\times\mathbb{P}^{2}$,
and denote the resulting scheme by $\widetilde{\mathcal{S}}$. Let
$\widetilde{\mathbf{D}}_{5}$ be the exceptional divisor of $\mathbf{D}_{5}$,
and let $\widetilde{\mathcal{S}}_{0}$ be the restriction of
$\widetilde{\mathcal{S}}$ over
$\widetilde{\mathbf{K}}\setminus\widetilde{\mathbf{D}}_{5}$. Finally, let us
denote $\overline{\mathcal{S}}$ as the _scheme-theoretic_ image of
$\widetilde{\mathcal{S}}_{0}$ in $\widetilde{\mathbf{K}}\times\mathbb{P}^{2}$.

$\textstyle{\overline{\mathcal{S}}:=\overline{\widetilde{\mathcal{S}}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathbf{K}}\times\mathbb{P}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathbf{D}}\times\mathbb{P}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi|_{\widetilde{\mathbf{D}}}\times\text{id}}$$\scriptstyle{\widetilde{j}\times{\text{id}}}$$\textstyle{\mathbf{D}\times\mathbb{P}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathbf{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\widetilde{\mathbf{D}}:=\mathbf{D}\times_{\mathbf{K}}\widetilde{\mathbf{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{j}}$$\scriptstyle{\pi|_{\widetilde{\mathbf{D}}}}$$\textstyle{\mathbf{K}}$$\textstyle{\mathbf{D}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$

Let $\widetilde{\mathcal{Z}}=(\pi\times\text{id})^{-1}(\mathcal{Z})$ be the
pull-back of the universal subscheme $\mathcal{Z}$ along the map
$\pi\times\text{id}$. Recall that $j:\mathbf{D}\hookrightarrow\mathbf{K}$ is
the embedding map (Lemma 4.2). Also $\mathcal{W}$ is the universal family of
single point in $\mathbb{P}^{2}$ parameterized by $\mathbf{D}$ (Remark 4.3).

######  Proposition 4.5.

Under the above notations and definitions, there exists an _nested_ inclusion

$\widetilde{\mathcal{Z}}\subset\overline{\mathcal{S}}\subset\widetilde{\mathcal{S}}\subset\widetilde{\mathbf{K}}\times\mathbb{P}^{2}$
(17)

such that $\overline{\mathcal{S}}$ is a flat family of subschemes in
$\mathbb{P}^{2}$ over $\widetilde{\mathbf{K}}$.

###### Proof.

The flat extension of subschemes can be regarded as a modification of sheaves
(or complexes). Let
$\widetilde{\mathcal{W}}:=(\pi|_{\widetilde{\mathbf{D}}}\times{\text{id}})^{*}(\mathcal{W})$.
In our setting, let

$\mathcal{G}=\text{Ker}\left\\{\mathcal{O}_{\widetilde{\mathcal{S}}}\oplus(\widetilde{j}\times{\text{id}})_{*}\mathcal{O}_{\widetilde{\mathcal{W}}}[1]\twoheadrightarrow\mathcal{O}_{\widetilde{\mathcal{S}}}|_{\widetilde{\mathbf{D}}_{5}\times\mathbb{P}^{2}}\oplus\mathcal{O}_{\widetilde{\mathcal{W}}}[1]\twoheadrightarrow\mathcal{O}_{\widetilde{\mathcal{S}}}|_{\widetilde{\mathbf{D}}_{5}\times\mathbb{P}^{2}}\right\\}$

be the elementary modification of the complex
$\mathcal{O}_{\widetilde{\mathcal{S}}}\oplus(\widetilde{j}\times{\text{id}})_{*}\mathcal{O}_{\widetilde{\mathcal{W}}}[1]$
along
$\widetilde{\mathbf{D}_{5}}\times\mathbb{P}^{2}\cong\widetilde{\mathbf{D}}\times\mathbb{P}^{2}$
by
$\mathcal{O}_{\widetilde{\mathcal{S}}}|_{\widetilde{\mathbf{D}}_{5}\times\mathbb{P}^{2}}$
([Lo13, Proposition 2.1]). The effect of the modification is to change the
sub/quotient sheaf. Specifically, for each closed point
$(q^{2},v)\in\widetilde{\mathbf{D}}_{5}=\mathbb{P}(\mathcal{N}_{\mathbf{D}_{5}|\mathbf{K}})$,
$[I_{q}]=[W_{2}^{*}=\langle x,y\rangle]\in\mathrm{Gr}(2,W_{3}^{*})$, the
modified complex fits into the following exact sequence:

$0\rightarrow\mathcal{O}_{q^{4}}\rightarrow\mathcal{G}|_{(p^{2},v)\times\mathbb{P}^{2}}\rightarrow\mathbb{C}_{q}[1]\rightarrow
0.$

We show that the complex $\mathcal{G}|_{(q^{2},v)\times\mathbb{P}^{2}}$ is
isomorphic to a non-split one (i.e., structure sheaf) by analyzing the
Kodaira-Spencer map as below.

$\textstyle{T_{q^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbf{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$KS$\textstyle{\mathrm{Ext}_{\mathbb{P}^{2}}^{1}(\mathcal{O}_{q^{4}}\oplus\mathbb{C}_{q}[1],\mathcal{O}_{q^{4}}\oplus\mathbb{C}_{q}[1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Ext}_{\mathbb{P}^{2}}^{1}(\mathbb{C}_{q}[1],\mathcal{O}_{q^{4}}\oplus\mathbb{C}_{q}[1])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{\mathbf{D}_{5}|\mathbf{K},q^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\text{KS}}}$$\textstyle{\mathrm{Ext}_{\mathbb{P}^{2}}^{1}(\mathbb{C}_{q}[1],\mathcal{O}_{q^{4}}).}$
(18)

One can easily see that the map KS in (18) descents to the normal space
$N_{\mathbf{D}_{5}|\mathbf{K},q^{2}}$. Also it induces a linear map
$\overline{\text{KS}}$ into

$\mathrm{Ext}_{\mathbb{P}^{2}}^{1}(\mathbb{C}_{q}[1],\mathcal{O}_{q^{4}})\cong\mathrm{Hom}_{\mathbb{P}^{2}}(\mathbb{C}_{q},\mathcal{O}_{q^{4}})\cong\mathrm{Ext}^{1}_{\mathbb{P}^{2}}(\mathbb{C}_{q},I_{q^{4}})\cong(\wedge^{2}W_{2}^{*})^{\otimes
2}\otimes\mathrm{Sym}^{3}W_{2}^{*},$

where the last isomorphism comes from the resolution

$0\longrightarrow(\wedge^{2}W_{2}^{*}\otimes\mathrm{Sym}^{3}W_{2}^{*})\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-5)\longrightarrow\mathrm{Sym}^{4}W_{2}^{*}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-4)\longrightarrow
I_{q}^{4}\longrightarrow 0$

of $I_{q}^{4}$ obtained by taking the fourth symmetric power of the Koszul
complex of the ideal sheaf $I_{q}$ (cf. Remark 4.3). However, according to
Lemma 4.6 below, the normal space $N_{\mathbf{D}_{5}|\mathbf{K},q^{2}}$ can be
identified with $\mathrm{Sym}^{3}W_{2}$. Since the diagram (18) is
$\mathrm{SL}(W_{2})$-equivariant and both spaces in the bottom row are
irreducible representations of $\mathrm{SL}_{2}$, it suffices to check that
the induced map $\overline{\text{KS}}$ is non-zero at a special point. For
instance, consider the curve-linear curve
$\mathcal{I}_{\mathcal{Z}_{t}}=\langle x^{2},xy,y^{2}+tzx\rangle$ approaching
the point $I_{q}^{2}$. Clearly for all
$t\in\mathbb{C}^{*}\subset\mathbf{K}\setminus\mathbf{D}_{5}$ and
$v=2\partial_{x}^{2}\partial_{y}\in\mathrm{Sym}^{3}W_{2}$. The image of $v$ by
$\overline{\text{KS}}$ is
$\overline{\text{KS}}(v)=\widetilde{\mathbb{C}[x,y,z]/J}$ for $J=\langle
x^{4},x^{3}y,x^{2}y^{2},xy^{3},y^{4}\rangle+\langle zx^{2}y\rangle$. Since
$I_{p}^{4}\subset J\subset I_{p}^{2}$, we have the inclusions of subschemes in
(17) by the existence of relative Quot scheme. ∎

######  Lemma 4.6.

The normal bundle of $\mathbf{D}_{5}$ in $\mathbf{K}$ is given by

$\mathcal{N}_{\mathbf{D}_{5}\mid\mathbf{K}}\cong\mathrm{Sym}^{3}\mathcal{D}^{*},$

where $\mathcal{D}$ is the universal subbundle of
$\mathbf{D}_{5}\cong\mathbf{D}=\mathrm{Gr}(2,W_{3}^{*})$.

###### Proof.

Let $[W_{2}^{*}]\in\mathrm{Gr}(2,W_{3}^{*})=\mathbf{D}\cong\mathbf{D}_{5}$
denote the point representing the subscheme defined by the ideal $\langle
x,y\rangle^{2}$. Since the space $\mathbf{K}$ around $\mathbf{D}_{5}$ is
isomorphic to $\mathbf{H}[3]$, we can apply the deformation theory of the
Hilbert scheme of points. It is well-known that

$\begin{split}T_{[\langle x,y\rangle^{2}]}\mathbf{K}&\cong T_{[\langle
x,y\rangle^{2}]}\mathbf{H}[3]\\\
&\cong\mathrm{Hom}_{\mathbb{C}[x,y,z]}(\langle
x^{2},xy,y^{2}\rangle,\mathbb{C}[x,y,z]/\langle
x^{2},xy,y^{2}\rangle).\end{split}$

Since $\mathbf{D}_{5}$ is $\mathrm{SL}_{3}$-transitively invariant, by varying
the space $W_{2}^{*}$, we get a canonical isomorphism

$\mathcal{T}_{\mathbf{K}}|_{\mathbf{D}_{5}}\cong\mathcal{H}om_{\mathbf{D}_{5}}(\mathrm{Sym}^{2}\mathcal{D},\mathcal{D}\otimes\mathcal{Q})\cong(\mathrm{Sym}^{2}\mathcal{D}^{*})\otimes(\mathcal{D}\otimes\mathcal{Q}).$

On the other hand, from the universal sequence
$0\rightarrow{\mathcal{D}}\rightarrow{W_{3}^{*}\otimes\mathcal{O}_{\mathbf{D}_{5}}}\rightarrow{\mathcal{Q}}\rightarrow
0$, we obtain $T_{\mathbf{D}_{5}}\cong\mathcal{D}^{*}\otimes\mathcal{Q}$. As
combining these ones, we can construct a commutative diagram

$\textstyle{\mathcal{T}_{\mathbf{D}_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{\mathbf{K}}|_{\mathbf{D}_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}_{\mathbf{D}_{5}\mid\mathbf{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}^{*}\otimes\mathcal{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{Sym}^{2}\mathcal{D}^{*}\otimes\mathcal{D})\otimes\mathcal{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sym}^{3}\mathcal{D}^{*}.}$

Here the below horizontal sequence is the exact sequence
$0\rightarrow{\mathcal{D}^{*}}\rightarrow{\mathrm{Sym}^{2}\mathcal{D}^{*}\otimes\mathcal{D}}\rightarrow{\mathrm{Sym}^{3}\mathcal{D}^{*}\otimes\mathcal{Q}^{*}}\rightarrow
0$ tensored with $\mathcal{Q}$, where the first map is given by

$\begin{split}\partial_{x}&\mapsto\partial_{x}^{2}\otimes
y-\partial_{x}\cdot\partial_{y}\otimes x,\\\
\partial_{y}&\mapsto\partial_{y}^{2}\otimes
x-\partial_{y}\cdot\partial_{x}\otimes y\end{split}$

such that the dual of $x$ (resp. $y$) is denoted by $\partial_{x}$ (resp.
$\partial_{y}$). ∎

From the inclusion of the statement in Proposition 4.5, we have a commutative
diagram:

$\textstyle{\mathrm{Sym}^{2}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{\overline{\mathcal{S}}}(4)\subset\mathcal{I}_{\widetilde{\mathcal{Z}}}(4)}$$\textstyle{\mathcal{I}_{\widetilde{\mathcal{S}}}(4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

over $\widetilde{\mathbf{K}}\times\mathbb{P}^{2}$. By pushing down this
diagram along the projection map
$p:\widetilde{\mathbf{K}}\times\mathbb{P}^{2}\rightarrow\widetilde{\mathbf{K}}$,
we have the followings.

######  Theorem 4.7.

The direct image sheaf
$\mathcal{V}^{\prime}:=p_{*}\mathcal{I}_{\overline{\mathcal{S}}}(4)$ is
locally free. Furthermore, $\mathcal{V}^{\prime}$ fits into a short exact
sequence

$0\rightarrow{\mathcal{V}^{\prime}}\rightarrow{p_{*}\mathcal{I}_{\widetilde{\mathcal{Z}}}(4)=\pi^{*}\mathcal{U}}\rightarrow{\pi^{*}\mathcal{U}/\mathcal{V}^{\prime}:=\mathcal{Q}}\rightarrow
0$

where the quotient sheaf $\mathcal{Q}$ is also locally free. This implies the
existence of an injective morphism

$\mathbb{P}(\mathcal{V}^{\prime})\subset\mathbb{P}(\pi^{*}\mathcal{U})$

over $\widetilde{\mathbf{K}}$ and thus a morphism into $\mathbf{R}_{4}$.

###### Proof.

The locally freeness of $\mathcal{V}^{\prime}$ is due to the fact that the
family $\overline{\mathcal{S}}$ is flat over $\widetilde{\mathbf{K}}$ and
$R^{i}p_{*}\mathcal{I}_{\overline{\mathcal{S}}}(4)=0$ for all $i\geq 1$.
Moreover, from the nested ideal sequence
$0\rightarrow{\mathcal{I}_{\overline{\mathcal{S}}}(4)}\rightarrow{\mathcal{I}_{\widetilde{\mathcal{Z}}}(4)}\rightarrow{\mathcal{I}_{\widetilde{\mathcal{Z}}\mid\overline{\mathcal{S}}}(4)}\rightarrow
0$ ((17) in Proposition 4.5) and
$R^{i}p_{*}\mathcal{I}_{\widetilde{\mathcal{Z}}}(4)=0$ for all $i\geq 1$, the
quotient sheaf $\mathcal{Q}$ is locally free. ∎

To compute the Chow ring of $\mathbb{P}(\mathcal{V}^{\prime})$, we refer to
the following proposition.

######  Proposition 4.8.

There exists an exact sequence

$0\rightarrow{\mathcal{V}}\rightarrow{\mathcal{V}^{\prime}}\rightarrow{\mathcal{V}^{\prime}/\mathcal{V}=:\mathcal{A}}\rightarrow
0,$

where $\mathcal{A}$ is isomorphic to

$\mathcal{A}\cong\mathcal{O}_{\widetilde{\mathbf{D}}_{5}}(\widetilde{\mathbf{D}}_{5})\otimes\pi|_{\widetilde{\mathbf{D}}_{5}}^{*}\mathcal{O}_{\mathbf{D}_{5}}(-2)$

for the restriction map
$\pi|_{\widetilde{\mathbf{D}}_{5}}:\widetilde{\mathbf{D}}_{5}\rightarrow\mathbf{D}_{5}$
of the blow-up map $\pi$.

###### Proof.

Note that
$\text{Supp}(\mathcal{A})\subset\widetilde{j}(\widetilde{\mathbf{D}})=\widetilde{\mathbf{D}}_{5}$
for the inclusion map
$\widetilde{j}:\widetilde{\mathbf{D}}\hookrightarrow\widetilde{\mathbf{K}}$.
By pushing down the exact sequence
$\mathrm{Sym}^{2}\mathcal{F}\to\mathcal{I}_{\overline{S}}(4)\to\mathcal{I}_{\overline{S}|\widetilde{S}}(4)\to
0$ along the projection map
$p:\widetilde{\mathbf{K}}\times\mathbb{P}^{2}\to\widetilde{\mathbf{K}}$, we
obtain an exact sequence

$0\to\mathcal{V}\to\mathcal{V}^{\prime}\to
p_{*}\mathcal{I}_{\overline{S}|\widetilde{S}}(4),$

where the injection comes from the construction of locally free sheaves. On
the other hand, by Proposition 4.5, we obtain a commutative diagram

$\textstyle{\mathcal{O}_{\widetilde{\mathbf{K}}}(\widetilde{\mathbf{D}}_{5})|_{\widetilde{\mathbf{D}}_{5}}\otimes\mathcal{A}\cong\mathcal{T}or_{\widetilde{\mathbf{K}}}^{1}(\mathcal{A},\mathcal{O}_{\widetilde{\mathbf{D}}_{5}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{V}|_{\widetilde{\mathbf{D}}_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{V}^{\prime}|_{\widetilde{\mathbf{D}}_{5}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}|_{\widetilde{\mathbf{D}}_{5}}}$$\textstyle{\mathcal{O}_{\mathbf{D}}(-2)\cong\mathrm{Sym}^{2}(\wedge^{2}\mathcal{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sym}^{2}(\mathrm{Sym}^{2}(\mathcal{D}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sym}^{4}\mathcal{D},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the bottom row corresponds to the push-forward along $\widetilde{j}_{*}$
of the standard exact sequence of symmetric powers on
$\widetilde{\mathbf{D}}$. Thus we complete the proof. ∎

######  Corollary 4.9.

The bundle $\pi^{*}\mathcal{V}$ is the elementary transformation of
$\mathcal{V}^{\prime}$ by $\mathcal{A}$ along the divisor
$\widetilde{\mathbf{D}}_{5}$. Hence there exists a flip between
$\mathbb{P}(\pi^{*}\mathcal{V})$ and $\mathbb{P}(\mathcal{V}^{\prime})$ over
$\widetilde{\mathbf{K}}$ by the general result of Maruyama [Mar73].

From Theorem 4.7 and Corollary 4.9, we have a commutative diagram

${\mathbb{P}(\pi^{*}\mathcal{V})}$${\mathbb{P}(\mathcal{V}^{\prime})}$${\mathbb{P}(\mathcal{V})}$${\mathbb{P}(\mathcal{U})}$${\widetilde{\mathbf{K}}}$${\widetilde{\mathbf{K}}}$${\mathbf{K}}$${\mathbf{K}.}$flip$\scriptstyle{\pi}$$\scriptstyle{\pi}$

###  4.3. Description of the contraction map
$\mathbb{P}(\mathcal{V}^{\prime})\rightarrow\mathbf{R}_{4}$

In this subsection, we analyze the contraction of the induced map
$\mathbb{P}(\mathcal{V}^{\prime})\rightarrow\mathbf{R}_{4}\subset\mathbb{P}(\mathcal{U})$.

######  Proposition 4.10.

The birational morphism
$\mathbb{P}(\mathcal{V}^{\prime})\rightarrow\mathbf{R}_{4}$ contracts the
exceptional divisor (i.e., a $\mathbb{P}^{5}$-bundle over
$\widetilde{\mathbf{D}}_{5}$) to sublocus of $\mathbf{R}_{4}$, which is a
$\mathbb{P}^{8}$-fiberation over $\mathbb{P}^{2}\cong\mathbf{D}_{5}$ ((1) of
Proposition 2.8).

###### Proof.

Let us fix a point $[I_{q}^{2}]=[\langle x,y\rangle^{2}]\in\mathbf{D}_{5}$.
The exceptional divisor $\widetilde{\mathbf{D}}_{5}$ over $[I_{q}^{2}]$ has
the following description. Let $t_{0},t_{1},t_{2},t_{3}$ be the local
coordinate of the exceptional fiber $\mathbb{P}(\mathrm{Sym}^{3}W_{2})$ over
the point $[I_{q}^{2}]\in\mathbf{D}_{5}$. Also let
$u_{0},u_{1},u_{2},u_{3},u_{4}$ and $k$ be the coordinates of the fiber
$\mathbb{P}^{5}(\mathcal{V}^{\prime}|_{[I_{q}^{2},t_{i}]})$. Then the quartic
forms parameterized by the fiber
$\pi|_{\widetilde{\mathbf{D}}_{5}}^{-1}([I_{q}^{2}])$ are given by the first
row in

$\begin{split}u_{0}x^{4}+u_{1}x^{3}y+u_{2}x^{2}y^{2}+u_{3}xy^{3}+u_{4}y^{4}&+k(t_{0}zx^{3}+t_{1}zx^{2}y+t_{2}zxy^{2}+t_{3}zy^{3})\\\
=u_{0}x^{4}+u_{1}x^{3}y+u_{2}x^{2}y^{2}+u_{3}xy^{3}+u_{4}y^{4}&+u_{5}zx^{3}+u_{6}zx^{2}y+u_{7}zxy^{2}+u_{8}zy^{3}.\end{split}$
(19)

On the other hand, let $u_{0},u_{1},\cdots$, $u_{7}$ and $u_{8}$ be the
homogenous coordinates of the projective space $\mathbb{P}^{8}$, which
parameterizes the quartic forms in the second row of (19). Let
$\mathbb{P}^{4}\subset\mathbb{P}^{8}$ be a linear space defined by
$u_{5}=u_{6}=u_{7}=u_{8}=0$. Then the blow-up of $\mathbb{P}^{8}$ along the
linear subspace $\mathbb{P}^{4}$ is isomorphic to the closure of the graph of
the rational map $\mathbb{P}^{8}\dashrightarrow\mathbb{P}^{3}$,
$[u_{0}:u_{1}:...:u_{8}]\mapsto[u_{5}:u_{6}:u_{7}:u_{8}]$. That is, the
closure is the subvariety:

$\mathcal{C}:=\left\\{[u_{0}:u_{1}:...:u_{8}]\times[t_{0}:t_{1}:t_{2}:t_{3}]\;\mid\;\mathrm{rk}\begin{bmatrix}u_{5}&u_{6}&u_{7}&u_{8}\\\
t_{0}&t_{1}&t_{2}&t_{3}\end{bmatrix}=1\right\\}\subset\mathbb{P}^{8}\times\mathbb{P}^{3}.$

The projection $\mathcal{C}\rightarrow\mathbb{P}^{3}$ into the second
component has a projective bundle structure with the fiber $\mathbb{P}^{5}$,
which is exactly the fiber
$\pi|_{\widetilde{\mathbf{D}}_{5}}^{-1}([I_{q}^{2}])$. This justifies the
equality of (19) and describes the contraction map
$\widetilde{\mathbf{D}}_{5}\rightarrow\mathbf{R}_{4}$ such that the image is a
$\mathbb{P}^{8}$-fiberation over $[I_{q}^{2}]\in\mathbf{D}_{5}$. Note that the
fiber of $\mathcal{U}$ over $[I_{q}^{2}]$ is
$\mathbb{P}(\mathcal{U}_{[I_{q}^{2}]})=\mathbb{P}(\mathrm{H}^{0}(I_{q}^{2}(4)))$.
∎

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