# Asymptotic geometric Tevelev degrees of hypersurfaces

Carl Lian  Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den
Linden 6
10099 Berlin, Germany<EMAIL_ADDRESS>
###### Abstract.

Let $(C,p_{1},\ldots,p_{n})$ be a fixed general pointed curve and let
$(X,x_{1},\ldots,x_{n})$ be a smooth hypersurface of degree $e$ and dimension
$r$ with $n$ general points. We consider the problem of enumerating maps
$f:C\to X$ of degree $d$ (as measured in the ambient projective space) such
that $f(p_{i})=x_{i}$. When $e$ is small compared to $r$ and $d$ is large
compared to $g$, $e$, and $r$, these numbers have been computed first by
passing to a virtual count in Gromov-Witten theory obtained by Buch-
Pandharipande, and then by showing (in work of the author with Pandharipande)
that the virtual counts are enumerative via an analysis of boundary
contributions in the moduli space of stable maps. In this note, we give a
simpler computation via projective geometry.

##  1\. Introduction

###  1.1. Tevelev degrees

Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $r$, and
let $\beta\in H_{2}(X,\mathbb{Z})$ be a non-zero effective curve class. Fix
$g\geq 0$ and $n\geq 0$ in the stable range $2g-2+n>0$ so that the moduli
space of stable curves $\overline{\mathcal{M}}_{g,n}$ is non-empty.

Let $\overline{\mathcal{M}}_{g,n}(X,\beta)$ the moduli space of stable maps,
and let $\mathcal{M}_{g,n}(X,\beta)$ be the open subspace of maps with smooth
domain. We consider the canonical morphisms

$\displaystyle\tau$
$\displaystyle:\mathcal{M}_{g,n}(X,\beta)\to\mathcal{M}_{g,n}\times X^{n}$
$\displaystyle\overline{\tau}$
$\displaystyle:\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,n}\times
X^{n}$

remembering in the first factor the (stabilized) domain curve of a stable map,
and in the second factor the images of the marked points.

We will be concerned with the (virtual) degrees of the maps
$\tau,\overline{\tau}$, counting maps $f:C\to X$ of degree $\beta$ with
$f(p_{i})=x_{i}$, where $(C,p_{1},\ldots,p_{n})$ is a fixed general curve and
$x_{1},\ldots,x_{n}\in X$ are general points. The moduli space of stable maps
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ has virtual dimension equal to the
dimension of $\overline{\mathcal{M}}_{g,n}\times X^{n}$ if and only if

(1) $\int_{\beta}c_{1}(T_{X})=r(n+g-1)\,.$

If the dimension constraint (1) holds, then we expect to find a finite number
of maps in question. We will assume (1), often without mention, throughout
this paper.

If it is further the case that all dominating components of
$\mathcal{M}_{g,n}(X,\beta)$ are generically smooth of the expected dimension,
then we may define the geometric Tevelev degrees of $X$ to be the
corresponding degrees of $\tau$.

######  Definition 1.

Assume that (1) holds and that all components of $\mathcal{M}_{g,n}(X,\beta)$
dominating $\mathcal{M}_{g,n}\times X^{n}$ under the map

$\tau:\mathcal{M}_{g,n}(X,\beta)\to\mathcal{M}_{g,n}\times X^{n}$

are generically smooth of the expected dimension. Then, we define the
geometric Tevelev degree ${\mathsf{Tev}}^{X}_{g,\beta,n}\in\mathbb{Z}$ to be
the (set-theoretic) degree of $\tau$.

By definition, the geometric Tevelev degree ${\mathsf{Tev}}^{X}_{g,\beta,n}$
counts maps with smooth domain. This count, when defined, is a priori always a
non-negative integer because automorphism groups never appear in a general
fiber of $\tau$, see [13, Remark 4].

As is well-known, the components of $\mathcal{M}_{g,n}(X,\beta)$ can behave
wildly and in particular have the wrong dimension, in which case the geometric
Tevelev degrees may not be well-defined. On the other hand, upon passing to
$\overline{\mathcal{M}}_{g,n}(X,\beta)$, one gains access to a virtual
fundamental class living in the expected homological dimension, and its
pushforward by $\tau$ allows one to define virtual Tevelev degrees in greater
generality.

######  Definition 2.

Assume that (1) holds. The virtual Tevelev degree
${\mathsf{vTev}}^{X}_{g,\beta,n}\in\mathbb{Q}$ is defined by

$\overline{\tau}_{*}([\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\operatorname{vir}})={\mathsf{vTev}}^{X}_{g,\beta,n}\cdot[\overline{\mathcal{M}}_{g,n}\times
X^{n}]\in A^{0}(\overline{\mathcal{M}}_{g,n}\times X^{n}),$

where

$\overline{\tau}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,n}\times
X^{n}$

is as above.

While the enumeration of maps with fixed domain, viewed from various
perspectives, is an old question, the systematic study of Tevelev degrees has
received renewed interest after a result of Tevelev [15, Theorem 6.2] showing
that ${\mathsf{Tev}}^{\mathbb{P}^{1}}_{g,(g+1)[\mathbb{P}^{1}],g+3}=2^{g}$
(see Definition 1) for all $g\geq 0$. We review some of the subsequent
results.

###  1.2. Calculations of virtual Tevelev degrees

The virtual Tevelev degrees are, at least in principle, easier to compute,
owing to the following formula of Buch-Pandharipande.

######  Theorem 3.

[5, Theorem 1.3] Assume that (1) holds. Then, we have:

${\mathsf{vTev}}^{X}_{g,\beta,n}=\operatorname{Coeff}({\mathsf{P}}^{\star
n}\star{\mathsf{E}}^{\star g},q^{\beta}{\mathsf{P}}).$

Here, ${\mathsf{P}}\in H^{*}(X)\subset QH^{*}(X)$ denotes the point class in
the small quantum cohomology ring of $X$, ${\mathsf{E}}$ denotes the quantum
Euler class

${\mathsf{E}}=\sum_{j}\gamma_{j}^{\vee}\star\gamma_{j}\in QH^{*}(X)$

where $\\{\gamma_{j}\\},\\{\gamma^{\vee}_{j}\\}$ are dual bases for $H^{*}(X)$
with respect to the (classical) cup product, $\star$ denotes the quantum
product, and $\operatorname{Coeff}(\alpha,q^{\beta}{\mathsf{P}})$ denotes the
$q^{\beta}{\mathsf{P}}$-coefficient when $\alpha$ is expanded as a power
series in the quantum variable $q$.

From Theorem 3, one can obtain explicit formulas for virtual Tevelev degrees
in a number of examples. In the case $X=\mathbb{P}^{r}$, we have:

######  Theorem 4.

[5, (3)] Assume that (1) holds, that is, that $n=\frac{r+1}{r}\cdot d-g+1$.
Then,

${\mathsf{vTev}}^{\mathbb{P}^{r}}_{g,d,n}=(r+1)^{g}.$

Here, we use $\beta=d$ to denote the class of a curve of degree $d$. Note that
in order for $n=\frac{r+1}{r}\cdot d-g+1$ to be an integer, we require $r|d$.
A closely related virtual count was obtained by Bertram-Daskalopoulos-
Wentworth before the systematic development of virtual fundamental classes in
Gromov-Witten theory, see [3, Theorem 2.9].

For low degree hypersurfaces, there is a similar result. For a hypersurface
$X_{e}\subset\mathbb{P}^{r+1}$ of degree $e$, condition (1) becomes

(2) $n=\frac{r+2-e}{r}\cdot d-g+1.$

######  Theorem 5.

[5, Theorem 1.5] Let $X_{e}\subset\mathbb{P}^{r+1}$ be a smooth hypersurface
of degree $e$ and dimension $r$. Suppose that $3\leq e\leq\frac{r+3}{2}$.
Then, for all $g,d,n\geq 0$ satisfying $2g-2+n>0$ and (2), we have

${\mathsf{vTev}}^{X_{e}}_{g,d,n}=((e-1)!)^{n}\cdot(r+2-e)^{g}\cdot
e^{(d-n)e-g+1}.$

The case of quadric hypersurfaces $(e=2)$ is exceptional, see [5, Theorem
1.4].

A non-trivial aspect of the proof of Theorem 5 is controlling the
contributions from primitive cohomology of $X$. The explicit computation of
${\mathsf{vTev}}^{X_{e}}_{g,d,n}$ in the remainder of the Fano range
$\frac{r+4}{2}\leq e\leq r+1$ remains open, but Cela [6] has given partial
results.

See also [5, §3-4] for results on ${\mathsf{vTev}}^{X}_{g,\beta,n}$ when
$X=G/P$ (which also includes the case of quadric hypersurfaces).

###  1.3. Calculations of geometric Tevelev degrees

The computation of geometric Tevelev degrees is more subtle. Complete results
were first given in the case $X=\mathbb{P}^{1}$ by Cela-Pandharipande-Schmitt
[8] via intersection theory on Hurwitz spaces of branched covers. Brill-
Noether theory guarantees that the geometric Tevelev degrees of
$X=\mathbb{P}^{1}$ (in fact, of any $\mathbb{P}^{r}$) are well-defined for all
$d\geq 1$.

######  Theorem 6.

[8, Theorem 6] For all $g,d,n\geq 0$ satisfying $2g-2+n>0$ and $n=2d-g+1$, we
have:

${\mathsf{Tev}}^{\mathbb{P}^{1}}_{g,d,n}=2^{g}-\sum_{i=0}^{g-d-1}\binom{g}{i}+(g-d-1)\binom{g}{g-d}+(d-g-1)\binom{g}{g-d+1}.$

Binomial coefficients $\binom{g}{i}$ with $i<0$ are interpreted to vanish. In
particular, ${\mathsf{Tev}}^{\mathbb{P}^{1}}_{g,d,n}=2^{g}$ whenever $d\geq
g+1$ (equivalently, $d\leq n-2$), agreeing with the virtual count of Theorem
4.

Later work of the author with Farkas gave the following alternate formula via
limit linear series and Schubert calculus.

######  Theorem 7.

[9, Theorem 1.3] For all $g,d,n\geq 0$ satisfying $2g-2+n>0$ and $n=2d-g+1$,
we have:

${\mathsf{Tev}}^{\mathbb{P}^{1}}_{g,d,n}=\int_{\operatorname{Gr}(2,d+1)}\sigma_{1}^{g}\sum_{i+j=2d-2-g}\sigma_{i}\sigma_{j}.$

That the expression on the right evaluates to $2^{g}$ when $d\geq g+1$ is non-
trivial; a brute force calculation using the Pieri rule is possible, but a
combinatorial proof via the RSK algorithm was given by Gillespie-Reimer-Berg
[10]. The formulas of Theorems 6 and 7 have been extended to counts of covers
of $\mathbb{P}^{1}$ with arbitrary ramification profiles, see [7].

The calculation of ${\mathsf{Tev}}^{\mathbb{P}^{r}}_{g,d,n}$ for $r>1$ is
considerably more difficult and will be addressed in forthcoming work [12].
However, when $d$ is large compared to $r$ and $g$, it is again true that the
geometric and virtual Tevelev degrees agree:

######  Theorem 8.

[9, Theorem 1.2] Suppose that $d\geq rg+r$, or equivalently that $d\leq n-2$.
Then, for all $g\geq 0$, we have:

${\mathsf{Tev}}^{\mathbb{P}^{r}}_{g,d,n}=(r+1)^{g}.$

This leads to the following conjecture.

######  Conjecture 9.

Let $X$ be a smooth, projective Fano variety and let $g$ be a fixed genus.
Then, there exists a constant $C_{X,g}$ depending only on $X$ and $g$ such
that, if $\int_{\beta}c_{1}(T_{X})>C_{X,g}$, then

${\mathsf{vTev}}^{X}_{g,\beta,n}={\mathsf{Tev}}^{X}_{g,\beta,n}$

That is, the virtual Tevelev degree is enumerative when $\beta$ is
sufficiently positive. (In particular, the geometric Tevelev degree is well-
defined.)

Partial results in this direction have been given by the author with
Pandharipande:

######  Theorem 10.

Conjecture 9 holds for (among other examples):

  * •

[13, Theorem 10] homogeneous varieties $X=G/P$,

  * •

[13, Theorem 11] hypersurfaces $X_{e}\subset\mathbb{P}^{r+1}$ of degree $e$,
where $r>(e+1)(e-2)$.

Combining the enumerativity for hypersurfaces in Theorem 10 with the virtual
calculations of Theorem 5 yields an explicit asymptotic calculation of
geometric Tevelev degrees of hypersurfaces of low degree, that is, when the
degree $d$ of $f:C\to X$ is sufficiently large. We state the conclusion for
completeness.

######  Theorem 11.

Let $X_{e}\subset\mathbb{P}^{r+1}$ be a smooth hypersurface of degree $e$ and
dimension $r$. Suppose that $e\geq 3$ and $r>(e+1)(e-2)$. Then

${\mathsf{Tev}}^{X_{e}}_{g,d,n}=((e-1)!)^{n}\cdot(r+2-e)^{g}\cdot
e^{(d-n)e-g+1}$

_for all $d$ sufficiently large (compared to $g,e,r$)._

###  1.4. Geometric Tevelev degrees of hypersurfaces via projective geometry

Given the simplicity of the formulas of Theorem 11, the authors of [5]
speculate that a proof via projective geometry along the lines of the
asymptotic calculation of [13] for $X=\mathbb{P}^{r}$ should be possible (see
also [13, Question 28]). In this paper, we give such a proof.

We first outline the strategy; assume first for simplicity that $g=0$, where
the key idea already appears. Let $V=H^{0}(\mathbb{P}^{1},\mathcal{O}(d))$ and
let $\mathbb{P}(V^{r+2})\cong\mathbb{P}^{(r+2)(d+1)-1}$ denote the space of
$(r+2)$-tuples $[f_{0}:\cdots:f_{r+1}]$, where $f_{i}\in
H^{0}(\mathbb{P}^{1},\mathcal{O}(d))$, the $f_{j}$ taken up to simultaneous
scaling. A generic choice of $f_{j}$ defines a map
$f:\mathbb{P}^{1}\to\mathbb{P}^{r+1}$ of degree $d$, but the degree of this
map drops when the $f_{j}$ share common factors.

Now, fix general points $p_{1},\ldots,p_{n}\in\mathbb{P}^{1}$ and
$x_{1},\ldots,x_{n}\in X_{e}=Z(F)\subset\mathbb{P}^{r+1}$. The condition that
either $f(p_{i})=x_{i}$ or all of $f_{0},\ldots,f_{r+1}$ vanish at $p$ defines
a linear subspace ${\mathsf{Inc}}(p_{i},x_{i})\subset\mathbb{P}(V^{r+2})$ of
codimension $r+1$. We wish to intersect the ${\mathsf{Inc}}(p_{i},x_{i})$ with
the locus $J^{\prime}_{F}$ consisting of $f$ with image in $X_{e}$, that is,
with $F(f_{0},\ldots,f_{r+1})=0$. The subscheme $J^{\prime}_{F}$ is an
intersection of $de+1$ hypersurfaces of degree $e$ in $\mathbb{P}(V^{r+2})$.

One quickly sees that the loci ${\mathsf{Inc}}(p_{i},x_{i})$ and
$J^{\prime}_{F}$ do not intersect in the expected dimension, owing to the fact
that the $x_{i}$ are general points in $X_{e}$, not in $\mathbb{P}^{r+1}$.
This is easily circumvented by imposing instead the condition that
$f(p_{i})\in L_{i}$, where $L_{i}\subset\mathbb{P}^{r+1}$ is a fixed general
line through $x_{i}$; the corresponding subscheme
${\mathsf{Inc}}(p_{i},L_{i})\subset\mathbb{P}(V^{r+2})$ now has the correct
expected codimension of $r$ upon restriction to $J^{\prime}_{F}$, but
introduces an extra factor of $e$ in the intersection number, coming from the
$e-1$ other points of $L_{i}\cap X$.

There is, however, a more serious problem: typically, the intersection of the
${\mathsf{Inc}}(p_{i},L_{i})$ and $J^{\prime}_{F}$ contains positive-
dimensional excess loci corresponding to degenerate $f$. For example, if
$d\geq n$, we may take $f_{0},\ldots,f_{r+1}$ to be constant multiples of each
other vanishing on all of $p_{1},\ldots,p_{n}$, such that the image of the
constant map $f=[f_{0}:\cdots:f_{r+1}]$ lies in $X_{e}$. As (1) implies that

$n\sim\frac{r+2-e}{r}\cdot d,$

we have that if $e\geq 3$, even when $d$ becomes sufficiently large, such
degenerate maps $f$ always exist.

We resolve the issue by blowing up the linear subspaces on
$\mathbb{P}(V^{r+2})$ where all of $f_{0},\ldots,f_{r+1}$ vanish at one of the
$p_{i}$. More explicitly, let

$Y\subset\mathbb{P}(V^{r+2})\times(\mathbb{P}^{r+1})^{n}$

be the incidence correspondence (cut out by bilinear equations) consisting of
$(f,x^{\prime}_{1},\ldots,x^{\prime}_{n})$ for which either
$f_{0},\ldots,f_{r+1}$ simultaneously vanish on $p_{i}$ or
$f(p_{i})=x^{\prime}_{i}$. Then, we cut out (in a precise way) a subscheme
$J_{F}\subset Y$ approximating the _proper transform_
$J^{\prime}_{F}\subset\mathbb{P}(V^{r+2})$.

The key observation is that, on the locus where $f$ has a simple base-point
$p_{i}$, if $\widetilde{f}$ denotes the map of degree $d-1$ obtained by
dividing by the common factor, then
$(f,x^{\prime}_{1},\ldots,x^{\prime}_{n})\in J_{F}$ if and only if the points
$\widetilde{x}_{i}:=\widetilde{f}(p_{i})$ and $x^{\prime}_{i}$ lie on a line
contained in $X_{e}$ (Proposition 17).

This new condition may alternatively be regarded as the requirement that there
exist a stable map of degree $d$ to $X_{e}$ defined by the degree $d-1$ map
$\widetilde{f}:\mathbb{P}^{1}\to X_{e}$ and a rational tail mapping
isomorphically to the line on $X_{e}$ through $\widetilde{x}_{i}$ and
$x^{\prime}_{i}$. In light of this comparison, we employ a similar analysis as
in [13] to argue that, under the conditions of Theorem 11, the intersection of
$J_{F}$ and the pullback to $Y$ of

$\prod_{i=1}^{n}L_{i}\subset(\mathbb{P}^{r+1})^{n}$

is a reduced, zero-dimensional subscheme supported on the locus of $f$ which
are well-defined at all of the $p_{i}$. In particular, the degree of this
zero-dimensional subscheme is equal to
$e^{n}\cdot{\mathsf{Tev}}^{X_{e}}_{0,d,n}$.

When $g$ is arbitrary, we relativize this construction over the Jacobian
$\operatorname{Jac}^{d}(C)$ (following the strategy of [3]) to prove Theorem
11.

We will record, along the way of the proof, a new effective bound on $d$,
above which the virtual Tevelev degrees are enumerative. We now state the more
refined result.

######  Theorem 12.

Let $X_{e}\subset\mathbb{P}^{r+1}$ be a smooth hypersurface of degree $e$ and
dimension $r$. Suppose that $e\geq 3$ and $r>(e+1)(e-2)$.

Then,

${\mathsf{Tev}}^{X_{e}}_{g,d,n}=((e-1)!)^{n}\cdot(r+2-e)^{g}\cdot
e^{(d-n)e-g+1}$

for _all_ $d$ if $g=0$, and whenever

$d>\frac{r((3g-2)(1+e)+1+g(r+2))}{r-(e+1)(e-2)}$

if $g>0$.

That no condition on $d$ is necessary when $g=0$ is also [13, Corollary 34].

In fact, for a general hypersurface $X_{e}$ of degree $e\leq r-1$, then the
irreducibility of the full Kontsevich space $\overline{M}_{0,n}(X_{e},d)$ (due
to Riedl-Yang [14, Theorem 3.3], and also Harris-Roth-Starr [11], Beheshti-
Kumar [2] in smaller ranges for $e$) implies that the virtual Tevelev degree
${\mathsf{vTev}}^{X_{e}}_{0,d,n}$ is enumerative. In particular, if $3\leq
e\leq\frac{e+3}{2}$ and $X_{e}$ is general, then the formula
${\mathsf{Tev}}^{X_{e}}_{0,d,n}=((e-1)!)^{n}\cdot e^{(d-n)e+1}$ holds for a
fixed pair of integers $(d,n)$ satisfying (2), see Remark 23. (To guarantee
the formula to hold on $X_{e}$ for any $(d,n)$, one should take $X_{e}$ to be
very general.) We thank Eric Riedl for pointing this out.

We also note here that, while we will not need the arguments of [13] in full
generality, the transversality analysis of our new construction confronts the
same issues of controlling excess dimensions of curves on hypersurfaces, and
in particular, produces the same bound $r>(e+1)(e-2)$ for (asymptotic)
enumerativity. Thus, our proof of Theorem 12, which was previously established
through the virtual calculation of [5] and then the enumerativity analysis of
[13], may be viewed in comparison in the following light. First, we give a
more elementary path to the counts obtained as virtual degrees in [5]. Then,
we apply special cases of the arguments of [13] to show that these counts are
equal to the desired geometric Tevelev degrees.

###  1.5. Further directions

We make several remarks concerning extensions of this work.

  1. (i)

We expect the method to go through for complete intersections of low degree,
see also [5, §5] and [13, Proposition 26].

  2. (ii)

We will see that our calculation uses the fact that the degree $e$ of the
hypersurface $X_{e}$ is at least 3 in an essential way, see step (3) of §2.3.
However, we believe that minor modifications should allow one to handle the
exceptional case $e=2$ similarly.

  3. (iii)

Our transversality analysis in §2.4, which makes no improvements on the method
of [13], seems to go through highly suboptimal bounds. We expect more refined
estimates of excess dimensions of families of stable maps on hypersurfaces to
allow for stronger results.

  4. (iv)

While our calculation (provably) produces a geometrically enumerative count
only under the conditions given in Theorem 12, our construction produces a
cycle class of homological dimension 0 and degree equal to the corresponding
virtual Tevelev degree under the much milder bounds $r\geq 2e-3$ and $d\geq
2g$. Thus, it seems plausible that excess contributions in our geometric setup
can be identified with boundary contributions to the virtual Tevelev degrees
in greater generality.

  5. (v)

When $e>\frac{r+3}{2}$, our method fails because because the construction
imposes no non-trivial condition on $f$ whenever $f$ has a _double_ base-point
at $x_{i}$, which allows $f$ to vanish to order 2 at all $x_{i}$ whenever
$d\geq 2n$. However, working on further (iterated) blowups of
$\mathbb{P}(V^{r+2})$ should allow one to extend the calculation to larger
degrees $e$ and predict formulas for the corresponding virtual Tevelev
degrees, see [6].

  6. (vi)

One can consider more general insertions at the marked points of $X$, which
will be carried out in a later version of [5], see also [4]. More precisely,
for $i=1,2,\ldots,n$, fix a cohomology class $\Omega_{i}\in H^{m_{i}}(X)$.
Then, define ${\langle\Omega_{1},\ldots,\Omega_{n}\rangle}^{X}_{g,n}$ by the
integral

$\int_{\overline{\mathcal{M}}_{g,n}\times
X^{n}}\overline{\tau}_{*}([\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\operatorname{vir}})\cdot([(C,p_{1},\ldots,p_{n})]\otimes\Omega_{1}\otimes\cdots\otimes\Omega_{n})\in\mathbb{Q},$

where $[(C,p_{1},\ldots,p_{n})]\in A_{0}(\overline{\mathcal{M}}_{g,n})$ is the
class of any (general) curve and

$\overline{\tau}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,n}\times
X^{n}$

is the forgetful morphism as before. The case $\Omega_{i}=[x_{i}]\in
H^{2r}(X)$ corresponds to the virtual Tevelev degree
${\mathsf{vTev}}^{X}_{g,n,\beta}$ defined earlier. Note that
${\langle\Omega_{1},\ldots,\Omega_{n}\rangle}^{X}_{g,n}$ vanishes unless

$\int_{\beta}c_{1}(T_{X})=r(g-1)+\frac{1}{2}\sum_{i=1}^{n}m_{i}.$

When $X=X_{e}\subset\mathbb{P}^{r+1}$ is a smooth hypersurface and the
$\Omega_{i}$ are pullbacks of powers of the hyperplane class on
$\mathbb{P}^{r+1}$, our geometric calculation produces more generally a zero-
dimensional cycle of degree equal (up to a factor of $e^{n}$, as in the case
$\Omega_{i}=[x_{i}]$) to
${\langle\Omega_{1},\ldots,\Omega_{n}\rangle}^{X}_{g,n}$, see §2.6. We expect
again these counts to be enumerative for fixed $X,g$ and $d$ large, but we do
not consider this aspect of the question here.

###  1.6. Acknowledgments

We thank Anders Buch, Alessio Cela, Gavril Farkas, Maria Gillespie, Rahul
Pandharipande, Eric Riedl, Johannes Schmitt, and Jason Starr for discussions
about Tevelev degrees in various contexts. We also thank the referee for a
thorough reading and for pointing out a gap in the main construction in an
earlier version, among numerous other helpful comments. This project was
completed with the support of an NSF postdoctoral fellowship, grant
DMS-2001976, and the MATH+ Incubator grant “Tevelev Degrees.”

##  2\. Proof of Theorem 12

###  2.1. Preliminaries

Let $X=Z(F)\subset\mathbb{P}^{r+1}$, where $F\in
H^{0}(\mathbb{P}^{r+1},\mathcal{O}(e))$, be a smooth hypersurface of degree
$e\geq 3$, and let $x_{1},\ldots,x_{n}\in X$ be general points. (We will drop
the subscript $e$ from the introduction denoting the degree of $X$ in the rest
of the paper.) Fix a general pointed curve $(C,p_{1},\ldots,p_{n})$ and
general lines $L_{1},\ldots,L_{n}\subset\mathbb{P}^{r+1}$ such that $L_{i}\ni
x_{i}$ for each $i$. In particular, we assume that the $L_{i}$ are pairwise
disjoint and all intersect $X$ transversely.

We assume throughout that $d\geq 2g$. It is straightforward to check that this
is implied by the lower bound on $d$ imposed by Theorem 12. Let
$\operatorname{Jac}^{d}(C)$ be the Jacobian of degree $d$ line bundles on $C$,
let $\pi:C\times\operatorname{Jac}^{d}(C)\to\operatorname{Jac}^{d}(C)$ be the
projection, and let $\mathcal{P}$ be the Poincaré line bundle on
$C\times\operatorname{Jac}^{d}(C)$. Now, define the vector bundle
$\mathcal{V}=\pi_{*}(\mathcal{P})$ of rank $d-g+1$ on
$\operatorname{Jac}^{d}(C)$.

Let $\nu:\mathbb{P}(\mathcal{V}^{\oplus r+2})\to\operatorname{Jac}^{d}(C)$ be
the projective bundle whose points parametrize $(r+2)$-tuples
$f=[f_{0}:\cdots:f_{r+1}]$ of sections of $H^{0}(C,\mathcal{L})$ up to
simultaneous scaling. (We will generally drop the reference to $\mathcal{L}$
in referring to a point of $\mathbb{P}(\mathcal{V}^{\oplus r+2})$.) Given such
an $f=[f_{0}:\cdots:f_{r+1}]$, we say that a point $p\in C$ is a base-point
for $f$ if $f_{0},\ldots,f_{r+1}$ all vanish at $p$, a double base-point if
they do so to order at least 2, and a simple base-point if $p$ is a base-point
that is not a double base-point.

We denote by $\widetilde{f}:C\to\mathbb{P}^{r+1}$ the morphism of degree at
most $d$ obtained from $f$ by twisting down all base-points of $f$, and
(abusively) by $f:C\dashrightarrow\mathbb{P}^{r+1}$ to be the rational map
given simply by $f=[f_{0}:\cdots:f_{r+1}]$, which is undefined at any base-
points of $f$.

###  2.2. Blowing up the locus of maps with base-points

We work on the ambient space $\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times\mathbb{P}^{r+1}$. Fix any point $p\in C$, and let
$Y_{p}\subset\mathbb{P}(\mathcal{V}^{\oplus r+2})\times\mathbb{P}^{r+1}$ be
the locus of $(f,y)$ where either $p$ is a base-point of $f$ or $f(p)=y$.
Alternatively, we have a linear projection map
$\operatorname{ev}:\mathbb{P}(\mathcal{V}^{\oplus
r+2})\dashrightarrow\mathbb{P}^{r+1}$ sending $f$ to $f(p)$, defined exactly
where $p$ is not a base-point of $f$. Then, under

$(\operatorname{ev},\operatorname{id}):\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times\mathbb{P}^{r+1}\dashrightarrow\mathbb{P}^{r+1}\times\mathbb{P}^{r+1},$

we have that $Y_{p}$ is the closure of the pre-image of the diagonal
$\Delta\subset\mathbb{P}^{r+1}\times\mathbb{P}^{r+1}$.

Upon composing with the projection to $\mathbb{P}(\mathcal{V}^{\oplus r+2})$,
we may identify $b:Y_{p}\to\mathbb{P}(\mathcal{V}^{\oplus r+2})$ with the
blowup of $\mathbb{P}(\mathcal{V}^{\oplus r+2})$ at the locus where
$f_{0},\ldots,f_{r+1}$ all vanish at $p$. Because we have assumed that $d\geq
2g$, this locus is the projective sub-bundle
$\mathbb{P}(\mathcal{V}(-p)^{\oplus r+2})\subset\mathbb{P}(\mathcal{V}^{\oplus
r+2})$ of $(r+2)$-tuples of sections of $H^{0}(C,\mathcal{L}(-p))$ over each
point $[\mathcal{L}]$ of $\operatorname{Pic}^{d}(C)$.

The exceptional divisor $E\subset Y_{p}$ is equal to
$\mathbb{P}(\mathcal{V}(-p)^{\oplus r+2})\times\mathbb{P}^{r+1}$. Let
$H,H_{p}\in\operatorname{Pic}(Y_{p})$ denote the pullbacks of the relative
hyperplane class on $\mathbb{P}(\mathcal{V}^{\oplus r+2})$ and the hyperplane
class of $\mathbb{P}^{r+1}$ to $Y_{p}$, respectively.

######  Lemma 13.

We have $[E]=H-H_{p}$ in $\operatorname{Pic}(Y_{p})$.

###### Proof.

As a Cartier divisor on $Y_{p}$, we may identify $E$ with the locus of
$([f_{0}:\cdots:f_{r+1}],[y_{0}:\cdots:y_{r+1}])\in Y_{p}$ where $f_{0}(p)=0$,
which has class $H$, minus the locus where $y_{0}=0$, which has class $H_{p}$.
∎

######  Definition 14.

For $k=0,1,\ldots,e$, we define
$I^{(k)}_{F,p}\subset\mathbb{P}(\mathcal{V}^{\oplus r+2})$ to be the locus of
$[f_{0}:\cdots:f_{r+1}]$ for which $F(f_{0},\ldots,f_{r+1})\in
H^{0}(C,\mathcal{L}^{\otimes e})$ vanishes to order at least $k$ at $p$.

We now construct inductively a chain of subschemes

$J^{(e)}_{F,p}\subset\cdots\subset J^{(0)}_{F,p}=Y_{p}$

where $J^{(k)}_{F,p}\subset b^{-1}(I^{(k)}_{F,p})$ may be regarded as the
"virtual proper transform" of $I^{(k)}_{F,p}$ under $b$. Assuming that
$J^{(k-1)}_{F,p}\subset b^{-1}(I^{(k-1)}_{F,p})$, the subscheme
$J^{(k)}_{F,p}\subset J^{(k-1)}_{F,p}$ is cut out by the vanishing of a
section of a certain line bundle.

We first cut out a subscheme $\overline{J}^{(k)}_{F,p}\subset J^{(k-1)}_{F,p}$
by the divisorial condition that $F$ (which, by assumption, vanishes to order
at least $k-1$ at $p$) vanishes to order $k$ at $p$. This amounts to the
vanishing of a section

$F^{(k)}\in\ker(H^{0}(C,\mathcal{L}^{\otimes e}|_{kp})\to
H^{0}(C,\mathcal{L}^{\otimes
e}|_{(k-1)p}))\cong\frac{H^{0}(C,\mathcal{L}^{\otimes
e}(-(k-1)p))}{H^{0}(C,\mathcal{L}^{\otimes e}(-kp))}$

obtained by restricting $F(f_{0},\ldots,f_{r+1})\in
H^{0}(C,\mathcal{L}^{\otimes e})$. (Here, by $kp$ we mean the subscheme of $C$
of multiplicity $k$ supported at $p$, and by $\mathcal{L}^{\otimes e}|_{kp}$
we mean $\mathcal{L}^{\otimes e}\otimes_{\mathcal{O}_{C}}\mathcal{O}_{kp}$.)

Globally, $\overline{J}^{(k)}_{F,p}$ is the vanishing locus of a canonical
section

$s^{(k)}_{F}\in
H^{0}\left(J^{(k-1)}_{F,p},b^{*}\left(\nu^{*}\pi_{*}\left(\frac{\mathcal{P}^{\otimes
e}(-(k-1)p)}{\mathcal{P}^{\otimes
e}(-kp)}\right)\otimes\mathcal{O}_{\mathbb{P}}(e)\right)\right).$

Note in particular that $\deg(\mathcal{L}^{\otimes e}(-kp))=ed-k\geq
e(d-1)>2g-2$ by assumption, so $\mathcal{P}^{\otimes e}(-(k-1)p)$ and
$\mathcal{P}^{\otimes e}(-kp)$ have vanishing higher cohomology on the fibers
of $\pi:C\times\operatorname{Jac}^{d}(C)\to\operatorname{Jac}^{d}(C)$.

We now express this vanishing locus in terms of local coordinates. Let $t$ be
an analytic local coordinate on $C$ near $p$, so that
$\mathcal{L}|_{\widehat{\mathcal{O}_{C,p}}}\cong\mathbb{C}[[t]]$
simultaneously for all $\mathcal{L}\in\operatorname{Jac}^{d}(C)$. More
precisely, $t\in\widehat{\mathcal{O}_{C,p}}$ is obtained by restriction from
an everywhere non-vanishing section of the trivial line bundle
$\mathcal{P}|_{\operatorname{Jac}^{d}(C)\times\widehat{\mathcal{O}_{C,p}}}$.

Then, write

$f_{j}=f_{j,0}+f_{j,1}t+f_{j,2}t^{2}+\cdots$

for the Taylor expansion of $f_{j}$ around $p$, so that

$F(f_{0},\ldots,f_{r+1})=s^{(1)}_{F}+s^{(2)}_{F}t^{1}+\cdots+s^{(k-1)}_{F}t^{k-2}+s^{(k)}_{F}t^{k-1}+\cdots.$

By assumption, the equations $s^{(1)}_{F},s^{(2)}_{F},\ldots,s^{(k-1)}_{F}$
(which are all degree $e$ polynomials in the $f_{j,\ell}$) all vanish on
$J^{(k-1)}_{F,p}$, and $\overline{J}^{(k)}_{F,p}=V(s^{(k)}_{F})$. By
construction, we have $\overline{J}^{(k)}_{F,p}\subset b^{-1}(I^{(k)}_{F,p})$.

Note also that in these local coordinates, the exceptional divisor $E$ is the
common vanishing locus of the equations $f_{j,0}$, and $s_{F}^{(k)}$ has total
degree at least $e+1-k$ in the $f_{j,0}$. Alternatively, on the principal open
subset $D_{+}(y_{\ell})$ of $\mathbb{P}^{r+1}$, we may substitute
$f_{j,0}=f_{\ell,0}\cdot\frac{y_{j}}{y_{\ell}}$ for $j=0,1,\ldots,r+1$, and
$f_{\ell,0}$ is a local equation cutting out the exceptional divisor $E$.
Then, the equation $s^{(k)}_{F}$ is divisible by $f_{\ell,0}^{e+1-k}$.

We therefore conclude that the section $s^{(k)}_{F}$ vanishes globally upon
restriction to the Cartier divisor $(e+1-k)E$, and thus is the image of a
canonical section

$\widetilde{s}^{(k)}_{F}\in
H^{0}\left(J^{(k-1)}_{F,p},b^{*}\left(\nu^{*}\pi_{*}\left(\frac{\mathcal{P}^{\otimes
e}(-(k-1)p)}{\mathcal{P}^{\otimes
e}(-kp)}\right)\otimes\mathcal{O}_{\mathbb{P}(\mathcal{V}^{\oplus
r+2})}(e)\right)\otimes\mathcal{O}_{Y_{p}}(-(e+1-k)E)\right).$

Finally:

######  Definition 15.

Define $J^{(k)}_{F,p}\subset J^{(k-1)}_{F,p}$ by $V(\widetilde{s}^{(k)}_{F})$.

Clearly, $J^{(k)}_{F,p}\subset\overline{J}^{(k)}_{F,p}\subset
b^{-1}(I^{(k)}_{F,p})$. Locally, the section $\widetilde{s}^{(k)}_{F}$ is
given by $s^{(k)}_{F}/f_{\ell,0}^{e+1-k}$ on $D_{+}(y_{\ell})$ as above. We do
not make the claim that $J^{(k)}_{F,p}$ does not have a component supported on
$E$, so $J^{(k)}_{F,p}$ may be strictly larger than the proper transform of
$I^{(k)}_{F,p}$. Furthermore, we do not claim that $J^{(k)}_{F,p}$ has the
expected codimension of $k$ in $Y_{p}$.

However, as a result of the construction, we have a canonical excess class
$[J^{(e)}_{F,p}]^{\operatorname{exc}}\in A_{\dim(Y_{p})-e}(J^{(e)}_{F,p})$
with the property that, upon restriction of the construction to any open
neighborhood of $Y_{p}$ in which $J^{(e)}_{F,p}$ is pure of the expected
codimension of $e$, we recover the fundamental class. Namely,

######  Lemma 16.

We have

$[J^{(e)}_{F,p}]^{\operatorname{exc}}=\left(\prod_{k=1}^{e}((k-1)H+(e+1-k)H_{p})\right)\cap[Y_{p}]$

in $A_{\dim(Y_{p})-e}(J^{(e)}_{F,p})$.

###### Proof.

$J^{(e)}_{F,p}\subset Y_{p}$ is cut out successively by the vanishing of
sections of the line bundles

$b^{*}\left(\nu^{*}\pi_{*}\left(\frac{\mathcal{P}^{\otimes
e}(-(k-1)p)}{\mathcal{P}^{\otimes
e}(-kp)}\right)\otimes\mathcal{O}_{\mathbb{P}(\mathcal{V}^{\oplus
r+2})}(e)\right)\otimes\mathcal{O}_{Y_{p}}(-(e+1-k)E)$

for $k=1,2,\ldots,e$. The claim now follows from Lemma 13 and the triviality
of the line bundle $\pi_{*}\left(\frac{\mathcal{P}^{\otimes
e}(-(k-1)p)}{\mathcal{P}^{\otimes e}(-kp)}\right)$. ∎

###  2.3. Main setup

We now set up the intersection theory calculation computing
${\mathsf{Tev}}^{X}_{g,d,n}$. In the first two steps, we repeat the
construction of §2.2 for the marked points $p_{1},\ldots,p_{n}$ taken
together. Then, we impose the condition that our map $f:C\to\mathbb{P}^{r+1}$
lie on $X$ globally (as opposed to “to order $e$" at the $p_{i}$). Finally, we
impose the incidence conditions at the marked points.

  1. (1)

Let $Y\subset\mathbb{P}(\mathcal{V}^{\oplus r+2})\times(\mathbb{P}^{r+1})^{n}$
be the intersection of the loci $Y_{p_{i}}$ pulled back from each of the $n$
factors $(\mathbb{P}^{r+1})^{n}$, as defined in the previous section.
Explicitly, $Y$ is the locus of $(f,x^{\prime}_{1},\ldots,x^{\prime}_{n})$
where, for each $i$, either $p_{i}$ is a base-point of $f$ or
$f(p_{i})=x^{\prime}_{i}$.

  2. (2)

We define a closed subscheme $J^{(e)}_{F}\subset Y$ by iterating the
construction of §2.2 for $p_{1},\ldots,p_{n}$, one point at a time. More
precisely, in the first step, we first define $J^{(e)}_{F,p_{1}}\subset Y$ by
pulling back the construction of $J^{(e)}_{F,p_{1}}\subset Y_{p_{1}}$ to $Y$.
In the second, we carry out the construction of $J^{(e)}_{F,p_{2}}$ now on the
ambient space given by the pullback of $J^{(e)}_{F,p_{1}}$ to $Y$ instead of
on $Y_{p_{2}}$, cutting out the successive conditions that $F$ vanish to order
$k=1,2,\ldots,e$ at $p_{2}$, now having already imposed the additional
condition of $e$-order vanishing at $p_{1}$. There are $n$ steps in total.

At the $m$-th step of this construction, we use the fact that the line bundle
$\mathcal{L}^{\otimes e}(-ep_{1}-\cdots-ep_{m-1}-kp_{m})$ has degree at least

($*$) $e(d-n)=e\left(d\cdot\frac{e-2}{r}+g-1\right)>2g-2$

when $k\leq e$, when imposing the condition that a canonical section of
$\mathcal{L}^{\otimes e}(-ep_{1}-\cdots-ep_{m-1}-kp_{m}))$ vanishes to order
$k$ at $p_{m}$.

We obtain in the end a subscheme $J^{(e)}_{F}\subset Y$ of expected (but not
necessarily actual) codimension $ne$, equal to the intersection of the
pullbacks of the $J^{(e)}_{F,p_{j}}\subset Y_{p_{j}}$ to $Y$, and contained in
the locus of $(f,x^{\prime}_{1},\ldots,x^{\prime}_{n})$ for which
$F(f_{0},\ldots,f_{r+1})$ vanishes to order $e$ at all of the $p_{i}$. We also
get an excess cycle $[J^{(e)}_{F}]^{\operatorname{exc}}\in
A_{\dim(Y)-ne}(J^{(e)}_{F})$ in a similar way to before, from the first Chern
classes of the line bundles whose sections cut out $J^{(e)}_{F}$.

  3. (3)

Now, on $J^{(e)}_{F}$, because the section $F(f_{0},\ldots,f_{r+1})\in
H^{0}(C,\mathcal{L}^{\otimes e})$ vanishes to order $e$ at
$p_{1},\ldots,p_{n}$, it may be regarded naturally as a section of
$H^{0}(C,\mathcal{L}^{\otimes e}(-ep_{1}-\cdots-ep_{n}))$. We then define
$J_{F}\subset J^{(e)}_{F}$ by the vanishing of this section.

Globally, we have a canonical section

$s^{\prime}_{F}\in H^{0}(\mathbb{P}(\mathcal{V}^{\oplus
r+2}),\nu^{*}(\pi_{*}(\mathcal{P}^{\otimes e}(-ep_{1}-\cdots-
ep_{n}))\otimes\mathcal{O}_{\mathbb{P}}(e))),$

and $J_{F}\subset J^{(e)}_{F}$ is by definition equal to $V(s^{\prime}_{F})$.
We have applied here the inequality ($*$ ‣ (2)) to ensure that the line bundle
$\mathcal{P}^{\otimes e}(-ep_{1}-\cdots-ep_{n})$ has vanishing higher
cohomology on the fibers of $\pi$ (note that it is crucial here that $e\geq
3$). Note in particular that $F(f_{0},\ldots,f_{r+1})$ vanishes (now to every
order at every point) everywhere on $J_{F}$.

As in step (2), we do not assume that $V(s^{\prime}_{F})$ has the expected
codimension of $e(d-n)-g+1$. However, we get an excess class

$[J_{F}]^{\operatorname{exc}}:=c_{e(d-n)-g+1}(\nu^{*}(\pi_{*}(\mathcal{P}^{\otimes
e}(-ep_{1}-\cdots-
ep_{n}))\otimes\mathcal{O}_{\mathbb{P}}(e)))\cap[J^{(e)}_{F}]^{\operatorname{exc}}$

in $A_{\dim(Y)-(de-g+1)}(J_{F})$ agreeing with the usual fundamental class
upon restriction to any open neighborhood of $Y$ in which we have dimensional
transversality for both steps (2) and (3).

  4. (4)

Finally, we define $T\subset J_{F}$ by the condition that $x^{\prime}_{i}\in
L_{i}$ for all $i$; $T$ is cut out by $r$ linear conditions on each of the $n$
factors $\mathbb{P}^{r+1}$.

The expected codimension of $T\subset\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times(\mathbb{P}^{r+1})^{n}$ is

$n(r+1)+en+[e(d-n)-g+1]+rn=g+[(r+2)(d-g+1)-1]+(r+1)n,$

so the expected dimension of $T$ is 0. When
$(f,x^{\prime}_{1},\ldots,x^{\prime}_{n})\in T$ and $f$ has no base-points,
the degree $d$ morphism $f:C\to X$ has $f(p_{i})\in L_{i}\subset X$.

###  2.4. Transversality

We now analyze in more detail the construction above, to show that, under the
conditions of Theorem 12, $T$ is in fact a reduced zero-dimensional scheme
whose points correspond (up to a factor of $e^{n}$, owing to the extra
intersections of $L_{i}$ with $X$) to maps enumerated in
${\mathsf{Tev}}^{X}_{g,d,n}$. There are three steps:

  1. (i)

Describe the points of $T$ set-theoretically (Proposition 17)

  2. (ii)

Show, under the conditions of Theorem 12, that all points of $T$ correspond to
maps counted in ${\mathsf{Tev}}^{X}_{g,d,n}$ (Corollary 20)

  3. (iii)

Show that $T$ is reduced of dimension 0 at all such points (Proposition 21)

######  Proposition 17.

Suppose that $(f,x^{\prime}_{1},\ldots,x^{\prime}_{n})\in T$. Let
$\widetilde{f}:C\to\mathbb{P}^{r+1}$ be the map defined by $f$ as in §2.1,
obtained by twisting down base-points. Then, $\widetilde{f}$ has image in $X$,
and in addition satisfies the property that for all $i=1,2,\ldots,n$, one of
the following holds:

  1. (i)

$p_{i}$ is not a base-point of $f$ and $\widetilde{f}(p_{i})=x^{\prime}_{i}$,
or

  2. (ii)

$p_{i}$ is a simple base-point of $f$, and
$\widetilde{x}_{i}:=\widetilde{f}(p_{i})$ and $x^{\prime}_{i}$ lie on a line
contained in $X$ (or are equal), or

  3. (iii)

$p_{i}$ is a double base-point of $f$.

###### Proof.

That $f$ has image in $X$ is guaranteed by steps (2) and (3) of the
construction of $T$, as the section $F(f_{0},\ldots,f_{r+1})\in
H^{0}(C,\mathcal{L}^{\otimes e})$ is constrained to vanish. If $p_{i}$ is not
a base-point of $f$, then step (1) guarantees that
$\widetilde{f}(p_{i})=x^{\prime}_{i}$. Now, it remains to consider the case in
which $f$ has a simple base-point of $f$ at $p_{i}$.

We adopt the local coordinates from §2.2 above, that is, the construction of
$J^{(e)}_{F,p}\subset Y_{p}\subset\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times\mathbb{P}^{r+1}$, where we take $p=p_{i}$. If $f$ has a simple
base-point at $p$, that is, the $f_{j,0}$ vanish for $j=0,\ldots,r+1$, but the
$f_{j,1}$ are not all 0, then we have

$\widetilde{x}:=\widetilde{x}_{i}=[f_{0,1}:\cdots:f_{r+1,1}].$

Consider now the equations $\widetilde{s}^{(k)}_{F}$. As in the discussion
preceding Definition 15, we restrict to the principal open subset
$D_{+}(y_{\ell})$ of $\mathbb{P}^{r+1}$, where we may replace the equations
$f_{j,0}$ with $f_{\ell,0}\cdot\frac{y_{j}}{y_{\ell}}$, then divide the
equation $s^{(k)}_{F}$ by $f_{\ell,0}^{e+1-k}$ to obtain
$\widetilde{s}^{(k)}_{F}$. For convenience, we also multiply $s^{(k)}_{F}$ by
the unit $y_{\ell}^{e+1-k}$. Then, upon restriction to the locus
$f_{\ell,0}=0$ (that is, the exceptional divisor, where $f$ has a base-point
at $p$), the only surviving terms of the equations $\widetilde{s}^{(k)}_{F}$,
for $k=0,1,\ldots,e-1$, are the monomials given by $e-k$ factors of the form
$y_{j}$ and $k$ factors of the form $f_{j,1}$.

Now, for indeterminates $\alpha,\beta$, consider the expansion

$F(\alpha f_{0,1}+\beta y_{0},\ldots,\alpha f_{r+1,1}+\beta y_{r+1})$

as a polynomial of total degree $e$ in the $f_{j,1},y_{j}$, where we evaluate
the function $F$ at a point $\alpha\cdot\widetilde{x}+\beta\cdot
y\in\mathbb{P}^{r+1}$ lying on the line between $\widetilde{x}$ and
$x^{\prime}=y=[y_{0}:\cdots:y_{r+1}]$. Then, the sum of the terms of total
bidegree $(k,e-k)$ in the $f_{j,1},y_{j}$, respectively, is precisely
$\alpha^{k}\beta^{e-k}\widetilde{s}^{(k+1)}_{F}$, which vanishes for
$k=0,1,\ldots,e-1$ by the construction of $J^{(e)}_{F}$, and for $k=e$ by the
fact that $F_{f}=0$ on $J_{F}$.

In particular, any point on the line between $\widetilde{x}$ and $x^{\prime}$
vanishes on $F$, so we obtain the needed conclusion. ∎

We now analyze the “degenerate” points of $T$ parametrizing $f$ having base-
points. We will require the following lemma, cf. [13, Proof of Proposition
26].

######  Lemma 18.

Let $D$ be a smooth curve of genus $g$, and let $j:D\to X$ be a morphism of
degree $d\geq 0$. Then

$h^{1}(D,j^{*}T_{X})\leq de+1+g(r+2).$

###### Proof.

From the short exact sequence

$0\to T_{X}|_{D}\to
T_{\mathbb{P}^{r+1}}|_{D}\to\mathcal{O}_{\mathbb{P}^{r+1}}(e)|_{D}\to 0,$

we have

$h^{1}(D,j^{*}T_{X})\leq
h^{0}(D,\mathcal{O}_{\mathbb{P}^{r+1}}(e))+h^{1}(D,T_{\mathbb{P}^{r+1}}).$

To bound the first term, we have

$h^{0}(D,\mathcal{O}_{\mathbb{P}^{r+1}}(e))=de-g+1+h^{1}(D,\mathcal{O}_{\mathbb{P}^{r+1}}(e))\leq
de+1.$

For the second, the Euler sequence

$0\to\mathcal{O}_{\mathbb{P}^{r+1}}\to\mathcal{O}_{\mathbb{P}^{r+1}}(1)^{\oplus
r+2}\to T_{\mathbb{P}^{r+1}}\to 0$

shows that $h^{1}(D,T_{\mathbb{P}^{r+1}})\leq
h^{1}(D,\mathcal{O}_{\mathbb{P}^{r+1}}(1)^{\oplus r+2})\leq g(r+2)$. Combining
yields the claim. ∎

Suppose now that, for general choices of
$(C,p_{1},\ldots,p_{n})\in\mathcal{M}_{g,n}$ and $x_{i}\in L_{i}\subset X$,
there exists a point $(f,x^{\prime}_{1},\ldots,x^{\prime}_{n})\in T$ of the
form that $f$ has:

  * •

$b_{1}$ simple base-points at $p_{1},\ldots,p_{b_{1}}$,

  * •

$b_{2}$ double base-points at $p_{n-b_{2}+1},\ldots,p_{n}$, and

  * •

$b_{0}$ base-points (counted with multiplicity) outside of the $p_{i}$.

Assume that $b_{0},b_{1},b_{2}$ are not all 0, that is, $f$ fails to be base-
point-free.

By Lemma 17, $\widetilde{x}_{1},\ldots,\widetilde{x}_{b_{1}}$ may be joined by
lines on $X$ with points
$x^{\prime\prime}_{1},\ldots,x^{\prime\prime}_{b_{1}}$, respectively, where
$x^{\prime\prime}_{i}\in L_{i}\cap X$. We use this fact to construct a stable
map from a certain nodal curve to $X$. Namely, let $D$ be the union of a genus
$g$ spine
$(D_{\operatorname{sp}},p_{1},\ldots,p_{n})\cong(C,p_{1},\ldots,p_{n})$
containing the points $p_{i}$, with smooth rational tails
$R_{1},\ldots,R_{b_{1}}$ attached to $D_{\operatorname{sp}}$ at
$p_{1},\ldots,p_{b_{1}}$, respectively. Let $p^{\prime}_{i}\in R_{i}$ be an
arbitrary smooth point, for $i=1,2,\ldots,b_{1}$.

Then, we construct the stable map $[\overline{f}:D\to
X]\in\overline{\mathcal{M}}_{g,n-b_{2}}(X,d^{\prime})$ by mapping
$D_{\operatorname{sp}}$ to $X$ via $\widetilde{f}$, and by mapping each
$R_{i}$ isomorphically to the line between $\widetilde{f}(p_{i})$ and
$x^{\prime\prime}_{i}$, such that
$\overline{f}(p^{\prime}_{i})=x^{\prime\prime}_{i}$. Denote the degree of
$\overline{f}$ by $d^{\prime}=d-b_{0}-2b_{2}$.

The map $\overline{f}$ lives in a stratum

$\mathcal{M}_{\Gamma}\subset\overline{\mathcal{M}}_{g,n-b_{2}}(X,d^{\prime})$

parametrizing stable maps whose domain curve is homeomorphic to $D$. (When
$b_{1}=0$, we have that $\mathcal{M}_{\Gamma}$ is the open stratum of stable
maps with smooth domain.)

######  Proposition 19.

Let $\tau:\mathcal{M}_{\Gamma}\to\overline{\mathcal{M}}_{g,n-b_{2}}\times
X^{n-b_{2}}$ be the forgetful map. Assume that at least one of
$b_{0},b_{1},b_{2}$ is strictly positive. Then, under the assumptions of
Theorem 12, the map $\tau$ is not dominant.

###### Proof.

Let
$\delta=\operatorname{vdim}(\overline{\mathcal{M}}_{g,n}(X,d))=\dim(\overline{\mathcal{M}}_{g,n}\times
X^{n})$. Then, we have

$\dim(\overline{\mathcal{M}}_{g,n-b_{2}}\times X^{n-b_{2}})=\Delta-(r+1)b_{2}$

and

$\displaystyle\operatorname{vdim}(\mathcal{M}_{\Gamma})$
$\displaystyle=\operatorname{vdim}(\overline{\mathcal{M}}_{g,n-b_{2}}(X,d^{\prime}))-b_{1}$
$\displaystyle=\delta-(d-d^{\prime})(r+2-e)-b_{2}-b_{1}$
$\displaystyle\leq\delta-(2r+5-2e)b_{2}-(r+2-e)b_{0}-b_{1}.$

Let $\mathcal{M}_{\Gamma}^{\operatorname{dom}}$ be the union of the components
of $\mathcal{M}_{\Gamma}$ dominating the target of $\tau$. We now argue as in
[13, Proposition 22] to show that

$\dim(\mathcal{M}_{\Gamma}^{\operatorname{dom}})<\dim(\overline{\mathcal{M}}_{g,n-b_{2}}\times
X^{n-b_{2}}),$

which will immediately yield a contradiction.

If $n-b_{1}-b_{2}\geq\max(2g,1)$, then [13, Proposition 13, Lemma 16] implies
that
$\dim(\mathcal{M}_{\Gamma}^{\operatorname{dom}})=\operatorname{vdim}(\mathcal{M}_{\Gamma})$,
so we have

$\displaystyle\operatorname{vdim}(\mathcal{M}_{\Gamma})$
$\displaystyle=\delta-(2r+5-2e)b_{2}-(r+2-e)b_{0}-b_{1}$
$\displaystyle<\delta-(r+1)b_{2}$
$\displaystyle=\dim(\overline{\mathcal{M}}_{g,n-b_{2}}\times X^{n-b_{2}}),$

as long as $r\geq 2e-3$ and at least one of $b_{0},b_{1},b_{2}$ is strictly
positive.

Otherwise, suppose that $n-b_{1}-b_{2}\leq\max(2g-1,0)$. We claim first that

$\dim(\mathcal{M}_{\Gamma}^{\operatorname{dom}})\leq\operatorname{vdim}(\mathcal{M}_{\Gamma})+h^{1}(D_{\operatorname{sp}},\overline{f}^{*}T_{X}).$

Indeed, for a general stable map $\overline{f}$ in
$\mathcal{M}_{\Gamma}^{\operatorname{dom}}$, all components $R_{i}$ of $D$
other than $D_{\operatorname{sp}}$ are lines passing through a general point
of $X$, onto which the restriction of $T_{X}$ is therefore globally generated.
Indeed, we have $T_{X}|_{D}\cong\sum_{i=1}^{r}\mathcal{O}(a_{i})$ for some
integers $a_{i}$, and the $a_{i}$ must be non-negative in order for the points
$x_{i}=\overline{f}(p_{i})$ to deform to first order in all tangent
directions.

In particular, these lines move in a family of expected dimension ([13,
Proposition 13]), and the condition of passing through any fixed node locally
has the expected codimension of $r$. As a result, the excess dimension
$\dim(\mathcal{M}_{\Gamma}^{\operatorname{dom}})-\operatorname{vdim}(\mathcal{M}_{\Gamma})$
is bounded above by that of the stable map when restricted to
$D_{\operatorname{sp}}$, which is
$h^{1}(D_{\operatorname{sp}},\overline{f}^{*}T_{X})$.

We now have

$\displaystyle\dim(\mathcal{M}_{\Gamma}^{\operatorname{dom}})$
$\displaystyle\leq\operatorname{vdim}(\mathcal{M}_{\Gamma})+h^{1}(D_{\operatorname{sp}},\overline{f}^{*}T_{X})$
$\displaystyle\leq\operatorname{vdim}(\mathcal{M}_{\Gamma})+(d^{\prime}-b_{1})e+1+g(r+2)$
$\displaystyle\leq\operatorname{vdim}(\mathcal{M}_{\Gamma})+(d-n-
b_{2}+\max(2g-1,0))e+1+g(r+2)$
$\displaystyle=\delta-(2r+5-2e)b_{2}-(r+2-e)b_{0}-b_{1}$
$\displaystyle\qquad\qquad+(d-n-b_{2}+\max(2g-1,0))e+1+g(r+2)$
$\displaystyle\leq\delta-(2r+5-2e)b_{2}-(n-b_{2}-\max(2g-1,0)))$
$\displaystyle\qquad\qquad+(d-n-b_{2}+\max(2g-1,0))e+1+g(r+2)$
$\displaystyle=\delta-(2r+4-e)b_{2}-n+(d-n)e$
$\displaystyle\qquad\qquad+\max(2g-1,0)(1+e)+1+g(r+2)$
$\displaystyle\leq\dim(\overline{\mathcal{M}}_{g,n-b_{2}}\times
X^{n-b_{2}})-n+(d-n)e$ $\displaystyle\qquad\qquad+\max(2g-1,0)(1+e)+1+g(r+2)$
$\displaystyle=\dim(\overline{\mathcal{M}}_{g,n-b_{2}}\times
X^{n-b_{2}})+d\cdot\frac{(e+1)(e-2)-r}{r}$
$\displaystyle\qquad\qquad+[\max(2g-1,0)+(g-1)](1+e)+1+g(r+2)$
$\displaystyle<\dim(\overline{\mathcal{M}}_{g,n-b_{2}}\times X^{n-b_{2}}),$

as long as $(e+1)(e-2)<r$ and

$d>\frac{r((\max(2g-1,0)+(g-1))(1+e)+1+g(r+2))}{r-(e+1)(e-2)}.$

When $(e+1)(e-2)<r$ and $g=0$, the lower bound on $d$ is negative, so we
obtain the conclusion for all $d$; if $g>0$, the bound becomes

$d>\frac{r((3g-2)(1+e)+1+g(r+2))}{r-(e+1)(e-2)}.$

∎

######  Corollary 20.

Any $(f,x^{\prime}_{1},\ldots,x^{\prime}_{n})\in T$ has $f$ base-point free.

###### Proof.

Proposition 19 shows that $\mathcal{M}_{\Gamma}$ does not dominate the target
of the forgetful map

$\tau:\overline{\mathcal{M}}_{g,n-b_{2}}(X,d^{\prime})\to\overline{\mathcal{M}}_{g,n-b_{2}}\times
X^{n-b_{2}}.$

Thus, varying over all possible $\Gamma$, we conclude that for a fixed general
pointed curve $(C,p_{1},\ldots,p_{n})$, there exists a non-empty open set
$U\subset X^{n}$ with the property that, for any
$(x^{\prime\prime}_{1},\ldots,x^{\prime\prime}_{n})\in U$, there does not
exist a stable map $\overline{f}$ as above with
$(b_{0},b_{1},b_{2})\neq(0,0,0)$ and
$\overline{f}(p^{\prime}_{i})=x^{\prime\prime}_{i}$.

It now suffices to show that, for fixed $(x_{1},\ldots,x_{n})\in U$, a general
choice of lines $L_{i}\ni x_{i}$ (assumed transverse to $X$) has the property
that, for any choices of $x^{\prime\prime}_{i}\in L_{i}\cap X$, we have
$(x^{\prime\prime}_{1},\ldots,x^{\prime\prime}_{n})\in U$. Suppose that this
is not the case. Then, consider the (proper) map

$\lambda:X^{n}\backslash U\to\prod_{i}\mathbb{P}_{i},$

where $\mathbb{P}_{i}\cong\mathbb{P}^{r}$ is the space of lines through
$x_{i}$, sending $(x^{\prime\prime}_{1},\ldots,x^{\prime\prime}_{n})\in X^{n}$
to the lines $L_{i}$ through $x_{i},x^{\prime\prime}_{i}$. For dimensional
reasons, $\lambda$ cannot be dominant, and a general collection of
$(L_{1},\ldots,L_{n})$ in the complement of the image suffices. ∎

######  Proposition 21.

Under the assumptions of Theorem 12, the subscheme $T$ is reduced of the
expected dimension 0. In particular, the intermediate subschemes

$J_{F}\subset J^{(e)}_{F}\subset Y\subset\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times(\mathbb{P}^{r+1})^{n}$

have the expected dimension in a neighborhood of $T$.

###### Proof.

The second claim will follow from the first. Corollary 20 shows that every
point of $T$ corresponds to an honest map $f:C\to X$ of degree $d$ with
$f(p_{i})\in L_{i}$. Consider now a non-zero tangent vector $v$ of
$T\subset\mathbb{P}(\mathcal{V}^{\oplus r+2})\times(\mathbb{P}^{r+1})^{n}$. We
first observe that, because the $L_{i},X$ intersect transversely at the
$x_{i}$, and because $f(p_{i})$ has been constrained to lie on both $L_{i},X$,
we have that $v$ is trivial upon projection to the factor
$(\mathbb{P}^{r+1})^{n}$.

From here, one easily verifies that, in fact, $v$ defines a non-trivial
relative tangent vector at $[f]$ to the forgetful morphism

$\tau:\mathcal{M}_{g,n}(X,d)\to\mathcal{M}_{g,n}\times X^{n}.$

On the other hand, [13, Proposition 14] shows that, over some open subset
$V\subset X^{n}$ (for fixed $(C,p_{1},\ldots,p_{n})$), such a relative tangent
vector cannot exist when $n\geq\max(2g,1)$. This inequality is immediate when
$g=0$ and, as one easily checks, follows from the required lower bound on $d$
given in Theorem 12 when $g>0$. The same argument from the proof of Corollary
20 guarantees that, for a general choice of lines $L_{i}$, we have
$(x^{\prime\prime}_{1},\ldots,x^{\prime\prime}_{n})\in V$ for any
$x^{\prime\prime}_{i}\in L_{i}\cap X$. The conclusion follows. ∎

######  Corollary 22.

We have $\deg(T)=e^{n}\cdot{\mathsf{Tev}}^{X}_{g,d,n}$.

###### Proof.

Corollary 20 and Proposition 21 show on the one hand that, for a general
choice of lines $L_{i}$, the fibers of the map

$\tau:\mathcal{M}_{g,n}(X,d)\to\mathcal{M}_{g,n}\times X^{n}.$

over any of the $e^{n}$ points
$((C,p_{1},\ldots,p_{n}),x^{\prime\prime}_{1},\ldots,x^{\prime\prime}_{n})$
with $x^{\prime\prime}_{i}\in L_{i}\cap X$ are reduced of dimension 0 and
degree ${\mathsf{Tev}}^{X}_{g,d,n}$, and on the other hand that, the union of
these fibers is (scheme-theoretically) isomorphic to $T$ for the same choice
of the $L_{i}$. The conclusion follows. ∎

######  Remark 23.

Let $X\subset\mathbb{P}^{r+1}$ be a general hypersurface of degree $e\leq
r-1$. Then, the Kontsevich space $\overline{M}_{0,n}(X,d^{\prime})$ is
irreducible and generically smooth of the expected dimension for all positive
$d^{\prime}\leq d$ [11, 2, 14].

If we assume further that $3\leq e\leq\frac{r+3}{2}$, then the arguments of
Proposition 19 and Corollary 20 show that every point of $T$ comes from a
base-point-free $f$. Here, we use only that $f$ cannot have a double base-
point at all of the $p_{i}$, and do not require any estimates on
$h^{1}(D_{\operatorname{sp}},\overline{f}^{*}T_{X})$. Moreover, the arguments
of Proposition 21 and Corollary 22 go through as well. Thus, the degree of
$T$, as computed in the next section, gives the genus 0 geometric Tevelev
degrees of any general hypersurface $X$ (up to a factor of $e^{n}$).

###  2.5. Calculation

We now compute the degree of $T$. We do so by following the construction; in
each step, we obtain a new (excess) cycle class by capping with a class pulled
back from the ambient space $\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times(\mathbb{P}^{r+1})^{n}$. Let $H=c_{1}(\mathcal{O}_{\mathbb{P}}(1))$
be the relative hyperplane class on $\mathbb{P}(\mathcal{V}^{\oplus r+2})$,
and let $H_{1},\ldots,H_{n}$ be the hyperplane classes on the factors
$\mathbb{P}^{r+1}$.

  1. (1)

$Y\subset\mathbb{P}(\mathcal{V}^{\oplus r+2})\times(\mathbb{P}^{r+1})^{n}$ is
cut out by pulling back the class of the diagonal
$\mathbb{P}^{r+1}\subset\mathbb{P}^{r+1}\times\mathbb{P}^{r+1}$ under the
rational map $\operatorname{ev}:\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times(\mathbb{P}^{r+1})^{n}\dashrightarrow(\mathbb{P}^{r+1}\times\mathbb{P}^{r+1})^{n}$.
Because $\operatorname{ev}$ is given by relative linear projections and the
codimension of the base locus is strictly larger than that of the class being
pulled back, the class of the subscheme $Y$ is

$[Y]=\left(\prod_{i=1}^{n}(H^{r+1}+H^{r}H_{i}+\cdots+H_{i}^{r+1})\right)\cap[\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times(\mathbb{P}^{r+1})^{n}].$

  2. (2)

Arguing as in Lemma 16, we have

$[J^{(e)}_{F}]^{\operatorname{exc}}=\left(\prod_{i=1}^{n}\prod_{k=1}^{e}((k-1)H+(e+1-k)H_{i})\right)\cap[Y]$

in $A_{\dim(Y)-ne}(J^{(e)}_{F})$. Indeed, $J^{(e)}_{F}\subset Y$ is cut out by
sections of the (pulled back) line bundles

$\mathcal{O}_{\mathbb{P}(\mathcal{V}^{\oplus
r+2})}(e)\otimes\mathcal{O}_{Y}(-(e+1-k)E_{i})$

for $k=1,2,\ldots,e$ and exceptional divisors $E_{i}\subset Y_{p_{i}}$.

(Compare [5, §5.3].)

  3. (3)

Consider the top Chern class

$c_{t}(\nu^{*}(\pi_{*}(\mathcal{P}^{\otimes e}(-ep_{1}-\cdots-
ep_{n}))\otimes\mathcal{O}_{\mathbb{P}(\mathcal{V}^{\oplus r+2})}(e)))$

where $t=(d-n)e-g+1$ is the rank of the bundle $\pi_{*}(\mathcal{P}^{\otimes
e}(-ep_{1}-\cdots-ep_{n}))$ on $\operatorname{Jac}^{d}(C)$. This is equal to

$\sum_{m=0}^{g}c_{m}(\pi_{*}(\mathcal{P}^{\otimes e}(-ep_{1}-\cdots-
ep_{n})))\cdot(eH)^{t-m}$

where we have dropped the $\nu^{*}$ on the bundle pulled back from
$\operatorname{Jac}^{d}(C)$. Note that the Chern classes in the first factor
vanish above degree $\dim(\operatorname{Jac}^{d}(C))=g$.

Now, a standard computation (see e.g. [1, Chapter VIII, §2]) shows that

$c_{m}(\pi_{*}(\mathcal{P}^{\otimes e}(-ep_{1}-\cdots-
ep_{n})))=\frac{(-e^{2})^{m}}{m!}\cdot\theta^{m}\in
H^{2m}(\operatorname{Jac}^{d}(C)),$

where $\theta\in H^{2}(\operatorname{Jac}^{d}(C))$ is the class of the theta
divisor.

Therefore,

$[J_{F}]^{\operatorname{exc}}=\left(e^{t}\cdot\sum_{m=0}^{g}\frac{(-e)^{m}}{m!}\cdot\theta^{m}H^{t-m}\right)\cap[J^{(e)}_{F}]^{\operatorname{exc}}.$

  4. (4)

$T\subset J_{F}$ is cut out by $r$ linear conditions on each factor of
$\mathbb{P}^{r+1}$, and has the expected dimension of 0. Therefore,

$[T]=\left(\prod_{i=1}^{n}H_{i}^{r}\right)\cap[J_{F}]^{\operatorname{exc}}$

Note that this holds even if the intermediate schemes $J_{F},J^{(e)}_{F}$ have
the wrong dimension; restricting the construction to a neighborhood of $T$
avoiding all excess components yields simply the fundamental class of $T$ in
the end.

######  Proposition 24.

The degree of $T$ as a 0-cycle on $\mathbb{P}(\mathcal{V}^{\oplus
r+2})\times(\mathbb{P}^{r+1})^{n}$ is

$(e!)^{n}\cdot(r+2-e)^{g}\cdot e^{(d-n)e+1}.$

###### Proof.

Applying the projection formula, we need to multiply the formulas for the four
classes computed above, and extract all coefficients with degree exactly $r+1$
in each $H_{i}$. Because we automatically get $r$ factors of $H_{i}$ from step
(4) and one factor of $H_{i}$ from the term $(eH_{i})$ in step (2), we must
take the term $H^{r+1}$ in all $n$ factors in step (1), and the terms
$(e-1)H,(e-2)H,\ldots,H$ each $n$ times (in addition to the contributions from
$eH_{i}$) in step (2).

We therefore find that

$[T]=(H_{1}\cdots H_{n})^{r+1}\cdot(e)!^{n}\cdot e^{t}\cdot
H^{n(r+1)+(e-1)n}\cdot\sum_{m=0}^{g}\frac{(-e)^{m}}{m!}\cdot\theta^{m}H^{t-m},$

Pushing forward to $\operatorname{Jac}^{d}(C)$ and applying the projection
formula, we obtain

$\nu_{*}[T]=(e)!^{n}\cdot
e^{t}\cdot\sum_{m=0}^{g}\frac{(-e)^{m}}{m!}\cdot\theta^{m}\cdot
s_{g-m}(\mathcal{V}^{\oplus r+2}).$

Finally, [1, Chapter VIII, §2] (see also [3, Proof of Theorem 2.9]) shows that

$s_{g-m}(\mathcal{V}^{\oplus
r+2})=(r+2)^{g-m}\cdot\frac{\theta^{g-m}}{(g-m)!},$

so we conclude that

$\displaystyle\deg(T)$ $\displaystyle=(e)!^{n}\cdot
e^{t}\cdot\frac{\deg(\theta^{g})}{g!}\cdot\sum_{m=0}^{g}\binom{g}{m}(-e)^{m}(r+2)^{g-m}$
$\displaystyle=(e!)^{n}\cdot(r+2-e)^{g}\cdot e^{(d-n)e-g+1}.$

where we apply Poincaré’s formula [1, p. 25] and have substituted back
$t=(d-n)e-g+1$. ∎

We now conclude the main Theorem.

###### Proof of Theorem 12.

Combine Proposition 24 and Corollary 22. ∎

###  2.6. More general insertions

We observe here that one can just as easily replace the incidence condition in
step (4) that $x^{\prime}_{i}\in L_{i}$ with the condition that
$x^{\prime}_{i}$ lie in a linear space
$X_{i}\cong\mathbb{P}^{\ell_{i}}\subset\mathbb{P}^{r+1}$. In order for the
expected number of maps $f:C\to X$ of degree $d$ satisfying the property that
$f(p_{i})\in X_{i}$ to be finite, we require

$d(r+2-e)=r(n+g-1)-\sum_{i=1}^{n}\ell_{i}.$

The corresponding virtual count is
${\langle\Omega_{1},\ldots,\Omega_{n}\rangle}^{X}_{g,n}$, where
$\Omega_{i}=[X_{i}\cap X]$, see ((vi)) of §1.5.

One may construct a scheme $T_{\ell_{1},\ldots,\ell_{n}}$ of expected
dimension 0 as above, parametrizing maps $f:C\to X$ of degree $d$ with
$f(p_{i})\in X_{i}$. Write now

$(1+z+\cdots+z^{r+1})\cdot\prod_{j=0}^{e-1}(j+(e-j)z)=\alpha_{1}z+\cdots+\alpha_{e+r+1}z^{e+r+1},$

where $\alpha_{1},\ldots,\alpha_{e+r+1}\in\mathbb{Z}_{>0}$. If it happens that
$T_{\ell_{1},\ldots,\ell_{n}}$ has actual dimension 0, then proceeding as in
Proposition 24, we compute

$\deg(T_{\ell_{1},\ldots,\ell_{n}})=(r+2-e)^{g}\cdot
e^{(d-n)e-g+1}\cdot\prod_{i=1}^{n}\alpha_{\ell_{i}},$

which matches the virtual degree
${\langle\Omega_{1},\ldots,\Omega_{n}\rangle}^{X}_{g,n}$ upon dividing by
$e^{n}$ as in the proof of Theorem 11, see [4, pp. 21-23]. We do not consider
the question of transversality here.

## References

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# Contrastive Learning based Semantic Communication for Wireless Image
Transmission

Shunpu Tang, Qianqian Yang, Lisheng Fan, Xianfu Lei, Yansha Deng, and Arumugam
Nallanathan S. Tang and L. Fan are both with the School of Computer Science
and Cyber Engineering, Guangzhou University, Guangzhou, China (e-mail:
<EMAIL_ADDRESS>lsfan@gzhu.edu.cn). Q. Yang is with the College of
Information Science and Electronic Engineering, Zhejiang University, Hangzhou,
China (e-mail: qianqianyang20@zju.edu.cn).X. Lei is with the Institute of
Mobile Communications, Southwest Jiaotong University, Chengdu, China (e-mail:
xflei@home.swjtu.edu.cn).Y. Deng is with the Department of Engineering, Kings
College London, London, U.K (e-mail:yansha.deng@kcl.ac.uk)A. Nallanathan is
with the School of Electronic Engineering and Computer Science, Queen Mary
University of London, London, U.K (e-mail: a.nallanathan@qmul.ac.uk).

###### Abstract

Recently, semantic communication has been widely applied in wireless image
transmission systems as it can prioritize the preservation of meaningful
semantic information in images over the accuracy of transmitted symbols,
leading to improved communication efficiency. However, existing semantic
communication approaches still face limitations in achieving considerable
inference performance in downstream AI tasks like image recognition, or
balancing the inference performance with the quality of the reconstructed
image at the receiver. Therefore, this paper proposes a contrastive learning
(CL)-based semantic communication approach to overcome these limitations.
Specifically, we regard the image corruption during transmission as a form of
data augmentation in CL and leverage CL to reduce the semantic distance
between the original and the corrupted reconstruction while maintaining the
semantic distance among irrelevant images for better discrimination in
downstream tasks. Moreover, we design a two-stage training procedure and the
corresponding loss functions for jointly optimizing the semantic encoder and
decoder to achieve a good trade-off between the performance of image
recognition in the downstream task and reconstructed quality. Simulations are
finally conducted to demonstrate the superiority of the proposed method over
the competitive approaches. In particular, the proposed method can achieve up
to 56% accuracy gain on the CIFAR10 dataset when the bandwidth compression
ratio is 1/48.

###### Index Terms:

Semantic communication, image transmission, contrastive learning, joint
source-channel coding.

##  I Introduction

Recently, semantic communication has emerged as a promising approach for
efficient image transmission in wireless network, and attracted increasing
research interests from academic. Compared with the conventional communication
paradigm based on Shannon’s theory, semantic communication aims to prioritize
to preserve meaningful semantic information over focusing on the accuracy of
transmitted symbols, which can significantly reduce the amount of data to be
transmitted and improve the communication efficiency[1].

A major challenge in semantic communication for image transmission is to
effectively extract semantic information at the transmitter while accurately
reconstructing it at the receiver under limited communication conditions. To
overcome this issue, deep learning (DL) has been applied in semantic
communication, due to its growing success in semantics extraction and
reconstruction. In this direction, the authors in [2, 3] proposed a DL base
joint source-channel coding (DeepJSCC), where the encoder and decoder were
designed based on autoencoder and jointly optimized for semantic information
transmission to achieve a good image reconstruction quality. Moreover, a
collaborative training framework for semantic communication was proposed in
[4], where users could train their semantic encoder to improve the performance
of unknown downstream inference tasks. However, these approaches still suffer
from limitations in achieving considerable inference performance and making a
good trade-off between it and image reconstruction quality according to
communication conditions.

In essence, the semantic distance between two irrelevant images is large due
to different main semantic information, while the distance is small enough
between two nearly identical images sharing the same semantic information.
Motivated by this fact, we propose to integrate contrastive learning (CL) with
semantic communication and design a two-stage training procedure. The key
design in this framework is that we regard the image corruption that occurs
during transmission over a limited channel as a form of data augmentation
operation in [5] and propose the semantic contrastive coding to reduce the
semantic distance (a.k.a semantic similarity) between the original and
reconstructed image while maintain the semantic distance among irrelevant
images for better discrimination. Based on contrastive loss, we design the
two-stage training procedure and loss functions for jointly optimizing the
encoder and decoder, thereby achieving a good balance between the inference
performance in the downstream task and reconstruction quality. Simulations are
finally conducted to demonstrate the superiority of the proposed method over
the competitive approaches.

##  II System model

This paper investigates a semantic communication system for wireless image
transmission, where a convoloutional neural network (CNN) based semantic
encoder and decoder are deployed in the transmitter and receiver,
respectively. The semantic encoder is used to extract the semantic information
of input image $\bm{x}\in\mathbb{R}^{c\times h\times w}$ and directly realize
the non-linear mapping from semantic information into the $k$-dim complex-
valued vector $\tilde{\bm{s}}\in\mathbb{C}^{k}$, given by

$\tilde{\bm{s}}=\mathcal{E}_{\theta_{1}}(\bm{x}),$ (1)

where $\mathcal{E}_{\theta_{1}}(\cdot)$ represents the semantic encoding
operation with parameter $\theta_{1}$, and $c$, $h$, and $w$ denote the number
of channels, height, and width of the image, respectively. To simplify, we use
$n=c\times h\times w$ to stand for the dimension of $\bm{x}$. Typically, $k<n$
should be satisfied to meet the bandwidth constraint, and $k/n$ is referred to
as the bandwidth compression ratio. In particular, a large bandwidth
compression ratio indicates a favorable communication condition, whereas a
small one denotes a limited usage of bandwidth. In addition, a power
normalization layer[2] is used at the end of the semantic encoding network to
satisfy the average power constraint
$\frac{1}{k}\mathbb{E}[\bm{s}^{*}\bm{s}]\leq P$ at the transmitter, which can
be written as

$\bm{s}=\sqrt{kP}\frac{\tilde{\bm{s}}}{\sqrt{\tilde{\bm{s}}^{*}\tilde{\bm{s}}}},$
(2)

where $\bm{s}$ is the channel input signal that meets the power constraint,
and ∗ denotes the conjugate transpose. Next, $\bm{s}$ will be transmitted over
the additive white Gaussian (AWGN) channel, given by

$\hat{\bm{s}}=\bm{s}+\bm{\epsilon},$ (3)

where $\hat{\bm{s}}$ is the received signal, and
$\bm{\epsilon}\in\mathbb{C}^{k}$ denotes the independent and identically
distributed (IID) channel noise sample, which follows symmetric complex
Gaussian distribution $\mathcal{CN}(0,\sigma^{2}\bm{I})$ with zero mean and
variance $\sigma^{2}$.

Figure 1: Network architecture of the semantic encoder and decoder.

The semantic decoder deployed at the receiver will reconstruct the original
image $\hat{\bm{x}}\in\mathbb{R}^{c\times h\times w}$ from $\hat{\bm{s}}$
according to

$\hat{\bm{x}}=\mathcal{D}_{\theta_{2}}(\bm{\hat{s}}),$ (4)

where $\mathcal{D}_{\theta_{2}}(\cdot)$ is the semantic decoding operation
parameterized by $\theta_{2}$. Subsequently, $\hat{\bm{x}}$ will be used to
exert downstream task and obtain the inference results through the following
process

$\bm{f}_{\bm{x}}=\mathcal{F}^{b}_{\phi_{1}}(\hat{\bm{x}}),$ (5)

where $\mathcal{F}^{b}_{\phi_{1}}(\cdot)$ characterized by parameter
${\phi_{1}}$ denotes the feature extraction operation performed by the CNN
backbone of downstream task, and
$\bm{f}_{\bm{x}}=\\{\bm{f}^{(1)},\bm{f}^{(2)},\cdots\bm{f}^{(C)}\\}$ is the
output feature map with $C$ channels. The inference results $\hat{\bm{y}}$ can
be obtained by passed $\bm{f}_{\bm{x}}$ to the classifier
$\mathcal{F}^{cls}_{{\phi_{2}}}(\cdot)$ with parameter $\phi_{2}$, which can
be expressed as

$\hat{\bm{y}}=\mathcal{F}_{\phi_{2}}^{cls}(\bm{f}_{x}).$ (6)

Since maintaining semantic information in the reconstructed image is crucial
for the inference performance, especially when the channel bandwidth is
limited. Therefore, it is of vital importance to design the semantic encoder
and decoder, as well as the training procedure.

##  III Proposed Framework

Figure 2: Illustration of the proposed semantic contrastive coding.

In this section, we will introduce the proposed CL based semantic
communication framework. Specifically, we first present the architecture of
the semantic encoder and decoder, and then we will provide the details of
semantic contrastive coding and the training procedure.

###  III-A Architecture of Semantic Encoder and Decoder

The network architecture of the semantic encoder and decoder plays a critical
role in the extraction of semantic information. Therefore, we do not utilize
the straightforward approach of stacked convolutional layers in [2], as this
simple architecture lacks this ability. The architecture of the proposed
semantic encoder and decoder are presented in Fig. 1. The semantic encoder
comprises a $5\times 5$ head convolution, two down-sampling modules, and a
channel coding module. Each down-sampling module includes a basic block in
ResNet[6] (we refer to as ResBolck) for capturing the spatial feature of the
image, and a $4\times 4$ convolution with stride $2$ for down-sampling the
image. The channel coding module is used to mitigate channel corruption and
output the $k$-dim complex-valued channel input that satisfies the bandwidth
and power constraint.

Moreover, we adopt a symmetrical architecture in the decoder, which consists
of a $5\times 5$ head convolution, two up-sampling modules, and a re-coding
module. In the up-sampling module, ResBolcks are also used as in the encoder
and we adopt the Pixel-Shuffle technology[7] to up-sample the image, as it can
provide a more efficient computing paradigm and better reconstruction
performance compared to transposed convolution used in [2]. The re-coding
module consists of a $3\times 3$ convolution followed by the Sigmoid activated
function to generate the reconstructed image. Notably, the batch normalization
and parametric rectified linear unit (PReLU) activated function are followed
with all convolutions, if not specified.

###  III-B Semantic Contrastive Coding

The key design of semantic contrastive coding is inspired by the success of CL
which employs data augmentation to generate samples with similar vision
representation and minimizes the distance among them to pretrain the backbone.
We modify the CL process to adapt it for the semantic communication system.
Specifically, we replace the data augmentation with the process of wireless
transmission, as the image corruption that occurs during the transmission can
be viewed as a form of data augmentation, and the original image and
reconstructed one should keep a small semantic distance for an efficient
semantic communication system. Moreover, we utilize a pretrained backbone to
extract features and a learnable projection network to map these features into
the semantic space. In further, by incorporating the contrastive loss in
semantic space, we jointly optimize the semantic encoder and decoder rather
than pretraining the backbone in CL.

The details of the proposed semantic contrastive coding are shown in Fig. 2.
The process begins with the semantic encoding and decoding for a typical image
$\bm{x}$ in a training batch $\mathcal{B}$, where we can obtain the
reconstructed $\hat{\bm{x}}$. The backbone network
$\mathcal{F}^{b}_{\phi_{1}}(\cdot)$ is applied to $\bm{x}$ and $\hat{\bm{x}}$,
which generates the feature maps
$\bm{f}_{\bm{x}}=\mathcal{F}^{b}_{\phi_{1}}(\bm{x})$ and
$\bm{f}_{\hat{\bm{x}}}=\mathcal{F}^{b}_{\phi_{1}}(\hat{\bm{x}})$,
respectively. Next, a fully connected projection network
$\mathcal{P}_{\psi}(\cdot)$ with learnable parameter $\psi$ followed by a
normalization operation maps the features into a semantic space defined as a
hypersphere. During the training stage, $\mathcal{P}_{\psi}(\cdot)$ can be
updated to enhance the understanding of features, thereby learning the mapping
from features to semantics. Specifically, the projected results of
$\bm{f}_{\bm{x}}$ and $\bm{f}_{\hat{\bm{x}}}$ can be represented as
$\bm{q}_{\bm{x}}=\mathcal{P}_{\psi}({\bm{f}_{\bm{x}}})$ and
$\bm{v}_{+}=\mathcal{P}_{\psi}({\bm{f}_{\hat{\bm{x}}}})$, respectively, where
$\bm{q}_{\bm{x}}$ is referred to as the anchor, and $\bm{v}_{+}$ is called the
positive. We can use the cosine similarity between anchor and positive to
define the semantic distance between $\bm{x}$ and $\hat{\bm{x}}$.

For the remaining samples $\bm{m}\in\mathcal{B}/\\{\bm{x}\\}$ within training
batch $\mathcal{B}$, the same procedures will be followed. Specifically, we
can obtain the feature map $\bm{f}_{\bm{m}}=\mathcal{F}_{b}(\bm{m})$ by feding
$\bm{m}$ into the backbone network, and then project $\bm{f}_{\bm{m}}$ into
the semantic space using $\bm{v}_{\bm{m}}=\mathcal{P}_{{\psi}}(\bm{f}_{m})$,
where we refer $\bm{v}_{\bm{m}}$ to as the negative. Similar, the semantic
distance between $\bm{x}$ and $\bm{m}$ can be defined as the cosine similarity
between anchor and  negative. The objective of semantic contrastive coding is
to minimize the semantic distance between the original and reconstructed image
while maximizing the semantic distance among the original image and the
irreverent images. Therefore, we can use the InfoNCE function[5] to define the
semantic contrastive loss for the training batch $\mathcal{B}$, which can be
expressed as

$\mathcal{L}_{sem}=\mathbb{E}_{\bm{x}\in\mathcal{B}}\bigg{\\{}-\log\frac{\exp(\bm{q}_{\bm{x}}\cdot\bm{v}_{+}/\tau)}{\sum_{\bm{m}\in\mathcal{B}/\\{\bm{x}\\}}\exp(\bm{q}_{\bm{x}}\cdot\bm{v}_{m}/\tau)}\bigg{\\}},$
(7)

where $\tau>0$ is the temperature coefficient used to smooth the probability
distribution. Next, we will introduce how to take into account the semantic
contrastive coding and semantic contrastive loss to design the loss function
and training procedure.

###  III-C Loss Function and Training Procedure

Based on the semantic contrastive coding, we design a two-stage training
strategy to optimize the semantic encoder and decoder. The first stage is pre-
training, where we employ the semantic contrastive coding approach to train
the weights of encoder $\theta_{1}$, decoder $\theta_{2}$ and project network
$\psi$ simultaneously. However, it is difficult to achieve a fast convergence
speed when we only optimize semantic contrastive loss. To tackle this issue,
we combine it with the reconstructed loss between $\bm{x}$ and $\hat{\bm{x}}$,
since reducing the reconstructed loss can help improve the convergence speed
in the early training rounds. Specifically, we apply the mean square error
(MSE) function to evaluate the reconstruction loss for training batch
$\mathcal{B}$, which can be expressed as

$\mathcal{L}_{rec}=\mathbb{E}_{\bm{x}\in\mathcal{B}}\bigg{\\{}\frac{1}{n}||\bm{x}-\hat{\bm{x}}||^{2}_{2}\bigg{\\}}.$
(8)

Therefore, the loss function in the first training stage can be summarized as
the linear combination, given by

$L_{1}=\alpha_{1}\mathcal{L}_{rec}+(1-\alpha_{1})\mathcal{L}_{sem},$ (9)

where $\alpha_{1}\in[0,1]$ is a hyper-parameter that controls the trade-off
between the two part loss functions. For instance, we can set to $\alpha=k/n$
in the practical semantic communication system. In this context, the system
prioritizes the preservation of semantic information over the reconstructed
quality when the bandwidth compression is small. In contrast, as the bandwidth
compression increases, the system shifts its focus towards maintaining the
reconstructed quality.

In the second training stage, we aim to further optimize the performance of
the semantic communication system by jointly fine-tuning the encoder, decoder,
and classifier with a small learning rate to achieve considerable inference
performance and reconstructed image quality. One reason of fine-tuning the
classifier is that the weights of the backbone and classifier are typically
trained without considering channel corruption, which causes that the outputs
of the backbone network may undergo substantial changes when the reconstructed
images are inputted instead of the original images. This can result in a
performance degradation. Therefore, fine-tuning the classifier with the
semantic encoder and decoder can mitigate this issue and help enhance the
semantics transmission. The loss function of this stage can be expressed as

$L_{2}=\alpha_{2}\mathcal{L}_{rec}+(1-\alpha_{2})\mathcal{L}_{Task},$ (10)

where $\alpha_{2}\in[0,1]$ is a hyper-parameter like $\alpha_{1}$ and
$\mathcal{L}_{Task}$ is the loss function of the downstream task.
Specifically, when the downstream task is a classification problem, the cross-
entropy function can be employed to model the loss, given by

$\mathcal{L}_{Task}=\mathbb{E}_{\bm{x}\in\mathcal{B}}\bigg{\\{}-\frac{1}{N_{cls}}\sum_{i=1}^{N_{cls}}y_{i}\log(\hat{y}_{i})\bigg{\\}},$
(11)

where $y_{i}$ and $\hat{y}_{i}$ represent the ground-truth and the predicted
probability of the $i$-th class, respectively. Notation $N_{cls}$ denotes the
number of classes in the dataset.

##  IV Simulations

(a)

(b)

Figure 3: Performance on CIFAR-10 versus the bandwidth compression ratio with
SNR = 20dB.

In order to verify the effectiveness of the proposed framework, we conduct
experiments on CIFAR-10, which contains 60,000 32$\times$32 color images
divided into 10 classes. The training set comprises 50,000 images, while the
test set contains 10,000 images. A pre-trained ResNet-56 [6] is used as the
backbone network and classifier for downstream inference, while the structure
of the semantic encoder and decoder is shown in Fig. 1. The projection network
adopts a two-layer fully connected structure with an output dimension of 32.
The number of training epochs for the two stages is set to 200 and 100,
respectively, with a batch size of 128. Besides, we use the Adam optimizer
with a learning rate of 0.01 for the first pre-training stage and 0.0001 for
the second fine-tuning stage. These learning rates will be adjusted every 50
epochs with a decay factor of 0.5. We compare the proposed method with the
following DL-based semantic communication methods,

  * •

DeepJSCC [2]: DL-based source-channel joint coding that maps the original
input to the channel input through the structure of an autoencoder.

  * •

DeepJSCC-ft: an extension of DeepJSCC with the second stage fine-tuning
strategy of our proposed method, in which the encoder, decoder and classifier
are updated with a small learning rate.

  * •

DeepSC [4]: a DL-based semantic coding framework that trains the semantic
encoder and decoder with both semantic and observation losses to achieve
efficient semantic information transmission. For a fair comparison, we freeze
its encoder and decoder after training and retrained the classifier with a
small learning rate.

(a)

(b)

Figure 4: Performance on CIFAR-10 versus the bandwidth compression ratio with
SNR = 5dB. Figure 5: Visualization comparison of several methods on the Kodak
dataset, where SNR $=$ 20dB and bandwidth compression ratio is 1/48.

3(a) compares the accuracy performance of the proposed method, DeepJSCC,
DeepJSCC-ft, and DeepSC, where SNR is set to 20dB, and the bandwidth
compression ratio $k/n$ varies from 1/48 to 1/2.5. From this figure, we can
find that the proposed method consistently outperforms or matches the
competitive methods in accuracy. Specifically, when the compression ratio is
1/2.5, the proposed method, DeepJSCC, and DeepJSCC-ft achieve the test
accuracy of about 94%, whereas DeepSC performs poorly with the accuracy of
only 91%. As the bandwidth compression ratio decreases, the proposed method
still maintains a comparable accuracy performance. For instance, the proposed
method can achieve accuracy of 90.14% and 87.65% at bandwidth compression
ratios of 1/24 and 1/48, respectively, which outperforms DeepSC by about 2.5%
and 1% at the corresponding bandwidth compression ratios and also shows an
accuracy gain of up to 56% over DeepJSCC. These results indicate that the
proposed framework can effectively extract semantic information to meet the
requirements of downstream task and removes irrelevant redundant information
to ensure the semantic information can be successfully transmitted, especially
when channel bandwidth is limited.

3(b) presents the peak signal-to-ration (PSNR) of the proposed and the three
completing methods , where SNR is set to 20dB and the bandwidth compression
ratio varies from 1/48 to 1/2.5. As shown in the figure, we can find that as
the bandwidth compression ratio increases, the PSNRs of all methods get
improved. Although the proposed method sacrifices some image quality to
prioritize semantic information when the bandwidth compression ratio is low,
it can quickly catch up with DeepJSCC’s PSNR at higher compression ratios.
Specifically, the proposed method achieves a PSNR of 38.75dB, which is close
to the 39.27dB of DeepJSCC, and outperforms DeepSC with 36.91dB when the
bandwidth compression ratio is 1/2.5. These results indicate that the proposed
method can prioritize to transmit semantic information over irrelevant
background information to ensure the performance of downstream task in
bandwidth-limited scenarios, and meanwhile transmit enough background
information to obtain good image quality when the bandwidth is not a
bottleneck. This further demonstrates the effectiveness of the proposed
method.

4(a) and 4(b) present the performance comparison of several methods under poor
channel conditions in terms of accuracy and PSNR, respectively. Specifically,
both figures consider a low SNR of 5dB, and the bandwidth compression ratio
varies from 1/48 to 1/2.5. From 4(a), we can observe that the proposed methods
still shows the superiority in terms of accuracy compared to the three
competitive methods, indicating its robustness in low SNR scenarios. From
4(b), we can find the proposed framework can adaptively sacrifice the global
information to obtain comparable semantic performance when the bandwidth
compression ratio is low, and meanwhile obtain enough reconstructed quality in
terms of PSNR as bandwidth compression ratio decreased. These results in both
figures further verify the effectiveness and robustness of the proposed method
in low SNR scenarios.

We also provide the visualization comparison of several methods on the Kodak
dataset in Fig. 5, where the encoder and decoder are trained on the STL10
dataset, SNR is set to 20dB and the bandwidth compression ratio is 1/48. From
this figure, we can observe that the proposed can effectively remove redundant
background information and meanwhile preserve the main semantic information,
resulting in less image corruption in semantic regions (e.g., macaws and
rafters in this figures) compared to the competitive methods. Moreover, the
proposed method achieves similar PSNR and multi-scale structural similarity
(MS-SSIM) performance with DeepJSCC and DeepJSCC-ft, indicating the
effectiveness of the proposed method in reconstructing semantic information.
These results further demonstrate the superiority of the proposed method in
achieving leading accuracy in downstream tasks over the compared methods.

##  V Conclusion

In this paper, we proposed a CL-based semantic communication framework for
wireless image transmission. The framework incorporated semantic contrastive
coding and a two-stage training procedure to enhance the extraction of
semantic information, and in order toachieve a better trade-off between
reconstruction quality and the performance of downstream task. We evaluated
the effectiveness of the proposed methods through simulations on CIFAR-10 and
Kodak datasets, which simulated results demonstrated the superiority of our
approach over existing methods.

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