# Knowledge-aware Neural Collective Matrix Factorization for Cross-domain
Recommendation

Li Zhang1, Yan Ge 2, Jun Ma3, Jianmo Ni4 and Haiping Lu1 1Department of
Computer Science, University of Sheffield, Sheffield, United Kingdom2
Department of Computer Science, University of Bristol, Bristol, United
Kingdom3 Amazon Inc., Seattle, WA, USA4 Google Inc., USA1{lzhang72,
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>4
<EMAIL_ADDRESS>
(2022)

###### Abstract.

Cross-domain recommendation (CDR) can help customers find more satisfying
items in different domains. Existing CDR models mainly use common users or
mapping functions as bridges between domains but have very limited exploration
in fully utilizing extra knowledge across domains. In this paper, we propose
to incorporate the knowledge graph (KG) for CDR, which enables items in
different domains to share knowledge. To this end, we first construct a new
dataset AmazonKG4CDR from the Freebase KG and a subset (two domain pairs:
movies-music, movie-book) of Amazon Review Data. This new dataset facilitates
linking knowledge to bridge within- and cross-domain items for CDR. Then we
propose a new framework, KG-aware Neural Collective Matrix Factorization (KG-
NeuCMF), leveraging KG to enrich item representations. It first learns item
embeddings by graph convolutional autoencoder to capture both domain-specific
and domain-general knowledge from adjacent and higher-order neighbours in the
KG. Then, we maximize the mutual information between item embeddings learned
from the KG and user-item matrix to establish cross-domain relationships for
better CDR. Finally, we conduct extensive experiments on the newly constructed
dataset and demonstrate that our model significantly outperforms the best-
performing baselines.

Cross-domain recommendation, knowledge graph, graph autoencoder.

††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining; August
14-18, 2022; Washington DC Convention Center, USA.††booktitle: Proceedings of
the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD
’22), August 14-18, 2022, Washington, DC, USA††price: 15.00††isbn:
978-1-4503-XXXX-X/18/06††doi: XXXXXXX.XXXXXXX

##  1\. Introduction

Cross-domain recommendation (CDR) (Fernández-Tobías et al., 2012) is a
promising solution to the data sparsity problem in recommender systems.
Conventional single-target CDR models leverage information from a richer
(source) domain to improve the recommendation performance in a sparser
(target) domain (Hu et al., 2013; Yuan et al., 2019a; Berkovsky et al., 2007).
To improve performance in both domains, recent dual-target CDR models (Man et
al., 2017; Zhu et al., 2019; Li and Tuzhilin, 2020) are proposed, which
enables bidirectional transfer across domains with dual-learning mechanism
(Zhang et al., 2019; He et al., 2016).

Despite encouraging results from existing CDR models, several key issues
remain unsolved (Zhu et al., 2021). Firstly, current models, including the
dual-target ones, can not simultaneously improve the performance in both
source and target domains due to negative transfer (Pan and Yang, 2009). In
general, the knowledge learned from the sparser domain is less accurate than
that learned from the richer domain. Thus, the recommendation performance in
the richer domain tends to decline if the transfer direction is simply
inverted. Secondly, current CDR models mainly use common users (Man et al.,
2017; Zhu et al., 2019) or mapping functions (Li and Tuzhilin, 2020) to build
connections between domains. In real-life scenarios, relationships between
items within or across domains can characterize item-wise semantic relatedness
to help understand user-item interaction patterns (Wang et al., 2020).
However, current CDR models are inadequate in capturing such useful item-item
relationships.

Figure 1.  Knowledge graph is a natural bridge that connects items from
different domains. For example, “Lord of the Ring” in movies can get connected
with “Harry Potter” in books via related genre Fantasy. Such inter-domain
knowledge can reveal similar semantic relations among items from different
domains to further improve cross-domain recommendation. This paper constructs
a new dataset and proposes a new model to achieve this goal.

In this paper, we aim to address this gap by leveraging knowledge graph (KG),
a natural bridge for items from different domains (Wang et al., 2017). KGs can
benefit the CDR task in multiple ways (Wang et al., 2018). First, rich and
explicit connections among items in the KG can help improve the recommendation
performance in each domain, particularly the sparser domain. As shown in Fig
1, a user who has watched “Harry Potter and the Deathly Hallows” is very
likely to have interest in the movie “Fantastic Beasts and Where to Find Them”
(directed by the same director), which can be recommended with the assistance
of domain-specific knowledge in the KG. Second, domains often share some
domain-general information. For example, genre can characterize both book and
movie domains. “Lord of the Ring” (from movies), “Harry Potter” (from books)
can be closely connected in the KG via the related genre Fantasy. KGs provide
a natural bridge to build connections between domains. Leveraging such
information can help models understand target or source items by associating
rich semantic relatedness among items from different domains and further
improve recommendation performance.

To build KG-aware CDR, three unique technical challenges arise. (1) Though
several datasets exist for KG-aware single-domain recommendation, no publicly-
available dataset exists for KG-aware CDR. (2) To improve CDR, item (entity)
embeddings (representations) learned from the KG should contain both domain-
specific and domain-general information, which typically comes from different
hops of neighbors in KG. The second challenge is to model both adjacent and
higher-order relations in the item representation learning process. (3) Item
embeddings learned from the KG and those from the user-item interaction matrix
should be closely related, e.g., highly correlated, so that cross-domain
relationships can be effectively established. How to ensure this is the third
challenge we need to overcome.

To address the challenges above, we construct a new dataset for KG-aware CDR
and propose a novel KG-aware Neural Collective Matrix Factorization (KG-aware
NeuCMF) model. Firstly, we construct a new dataset named Amazon product
Knowledge Graph for CDR (AmazonKG4CDR) using a subset (movie, book, and music)
of the Amazon Review Data (2018) (Ni et al., 2019) and the Freebase KG (Chah,
2017; Bollacker et al., 2008). Then, we propose a two-step framework for KG-
aware CDR. 1) We train a shared autoencoder using a relational graph
convolutional network (RGCN) on the knowledge graph following a contrastive
learning-style (Kipf and Welling, 2016; Schlichtkrull et al., 2018). GCN-based
encoders learn a node’s embedding by aggregating information from its
neighbors via non-linear transformation and aggregation (Kipf and Welling,
2017a). Long-range node dependencies can be captured by stacking multiple GCN
layers to propagate information for multiple hops (Xu et al., 2018). This
enables capturing both domain-specific and domain-general information from
different hops of neighbors in the KG. 2) To establish cross-domain
relationships, the embeddings learned from KG should be highly coherent with
those from the user-item interaction matrix. Therefore, we incorporate the
mutual information (MI) estimation (Belghazi et al., 2018) into the neural
collective matrix factorization (NeuCMF) framework. This mechanism allows our
model to preserve both user-item interaction and KG information across items.
Finally, we conduct extensive experiments on our newly constructed datasets
and demonstrate that our model significantly outperforms the best-performing
baselines, with up to 21% (movie), 15.18% (music) improvement, in terms of the
mean absolute error (MAE) in movie-music domains recommendation.

In summary, our contributions are threefold:

  * •

We construct and leverage the knowledge graph for CDR task. To the best of our
knowledge, this is the first time to apply KG information for CDR.

  * •

We propose a two-step KG-aware NeuCMF framework for KG-aware CDR, which
enables learned item embeddings can capture both user-item interactions,
domain-general, domain-specific information from the KG.

  * •

We conduct extensive experiments on the newly constructed datasets.
Experimental results show that our proposed model can significantly outperform
most state-of-the-art CDR models.

Figure 2. KG construction for Amazon products.

##  2\. Related Work

###  2.1. Cross-Domain Recommendation

Different from conventional single-domain recommendation, CDR can leverage
information from source domain to improve the performance of target domain
(Berkovsky et al., 2007; Fernández-Tobías et al., 2012), namely single-target
CDR, which is a powerful tool to deal with the data sparsity problem. These
approaches extend the single-domain recommendation models by utilizing same
contents, such as tags, reviews (Fernández-Tobías and Cantador, 2014; Yuan et
al., 2019b), common items or users (Singh and Gordon, 2008; Hu et al., 2018;
Lian et al., 2017) as the bridge between and transfer information between
domains (Hu et al., 2013; Loni et al., 2014; Sahebi and Brusilovsky, 2015;
Sahebi and Walker, 2014).

The single-target CDR approaches only focus on how to leverage the source
domain to help improve the recommendation accuracy on the target one, but not
vice versa. Recently, dual-target CDR mothods (Man et al., 2017; Zhu et al.,
2019; Li and Tuzhilin, 2020) has been proposed to improve the performance on
both source and target domains simultaneously by leveraging dual-transfer
learning strategies (Zhang et al., 2019; He et al., 2016). However, as
referred to as Negative Transfer (Pan and Yang, 2009), this idea does not
work, because the knowledge learned from the sparser domain is less accurate
than that learned from the richer domain, thus the recommendation accuracy on
the richer domain is more likely to decline by simply and directly changing
the transfer direction. Therefore, dual target CDR demands novel and effective
solutions. None of the current CDR models can indeed improve the performance
on both domains simultaneously, and they are significantly hindered by limited
information and connections between two domains.

###  2.2. Knowledge Graph for Recommendation

In recent years, introducing recommendations with the KG as side information
has attracted considerable interest (Wang et al., 2018; Wang et al., 2017,
2019a). A KG is a heterogeneous graph, where nodes represent as entities,
edges represent relations between entities and a fact in KG is usually
represented in the form of a triple (head entity, relation, tail entity) (Wang
et al., 2017). KGs contain rich semantic relatedness among items and
incorporating KGs in RS can help explore the latent connections and provide
explanations for recommended items (Guo et al., 2020). Currently, KG-aware RS
models are only for the single-domain RS (Catherine and Cohen, 2016; Wang et
al., 2018; Tang et al., 2019; Zhao et al., 2019; Wang et al., 2017, 2019a).
While one bottleneck for CDR is lacking of connections between domains. since
KGs can naturally connect different domains, it would be promising by
incorporating KG in the user-item interaction matrix for better cross-domain
recommendation performance.

##  3\. KG-aware NeuCMF models

In this section, we present the technical details of our proposed CDR model,
KG-aware Neural CMF (KG-NeuCMF) that aims to improve the performance of CDR by
leveraging the KG. This section first introduces how to construct the
knowledge graph for items. Then we formulate the task and present our proposed
framework: KG-NeuCMF.

###  3.1. KG Construction for CDR

To develop a knowledge-aware CDR system, a key issue is how to obtain rich and
structured knowledge information for items. Existing research works use side
information from the original recommender system, such as tags and reviews. We
argue that the KG information will provide additional useful information to
the CDR task, since the intra-domain relationship among items can be captured.
In this paper, we present AmazonKG4CDR V1.0, a new dataset linking KG
information for CDR, which can be useful for researchers in the related areas
to explore possible approaches with the rich KG information.

We use the widely used dataset, Amazon Review Data (2018) (Ni et al., 2019),
covering various domains, from which we select a subset that includes two
domain pairs: movie-music, movie-book, which are being linked together through
a common user ID identifying the same user. On the KG side, we use the well-
known KG: Freebase (Bollacker et al., 2008). It stores facts by triples of the
form $<head><relation><tail>$. Since Freebase shut down its services, we use
its latest public version. We map items into Freebase entities via title
matching if there is a mapping available. Fig.2 shows the whole linkage
process. Since we only have item Asins (IDs of Amazon products), we need to
get items’ titles from the Amazon Review metadata
first111https://nijianmo.github.io/amazon/index.html. These titles are later
used to get KG entity IDs from The Knowledge Graph Search API, which are used
to extract the graph information from Freebase.

During the linkage process, we have dealt with several problems that will
affect the quality of the extract knowledge graph. First, the correctness of
the extracted KG entity IDs should be ensured. For example, a query is “Harry
Potter” (a book name), and returned results can be both movies and books. So,
we filter returned results by their type and name to ensure extracted IDs are
correct. To ensure the KG quality, we preprocess the extracted KG by filtering
out infrequent entities (e.g., lower than 10 in both datasets) and retaining
the relations appearing in at least 100 triplets.

###  3.2. Problem Statement

In this paper, we study the problem of KG-aware CDR. Formally, we are given
two domains, a source domain $\mathcal{S}$ (e.g., movie recommendation) and a
target domain $\mathcal{T}$ (e.g., book recommendation) that can be
represented as two user-item interaction matrices $\mathbf{R}_{\mathcal{S}}$
and $\mathbf{R}_{\mathcal{T}}$, where $r_{ui}=1$ indicates that user $u$
engages with item $i$, otherwise $r_{ui}=0$. In real online shopping platforms
(e.g., Amazon), users in domain $\mathcal{S}$ and domain $\mathcal{T}$ often
overlap, meaning that they have purchased items in both domains. The set of
users in both domains are shared, denoted by $\mathcal{U}$ (of size $m$ =
$\left|\mathcal{U}\right|$). In our setting, there is no overlap of items
between two domains and each item only belongs to one single domain. Denote
the set of items in $\mathcal{S}$ and $\mathcal{T}$ by
$\mathcal{I_{\mathcal{S}}}$ and $\mathcal{I_{T}}$ with size $n_{\mathcal{S}}$
= $\left|\mathcal{I_{S}}\right|$) and $n_{\mathcal{T}}$ =
$\left|\mathcal{I_{T}}\right|$ respectively. Additionally, we also have a
knowledge graph $\mathcal{G}$, a multi-relational graph, containing rich facts
about items. Each fact in the KG is represented as a triple (head
entity,relation,tail entity) (($h,r,t$)) (Wang et al., 2017). The KG can
represent large-scale information from multiple domains (Ehrlinger and Wöß,
2016). In recommendation scenarios, an item in the user-item interaction
matrix corresponds to an entity in the KG.

Given $\mathbf{R}_{\mathcal{S}}$ and $\mathbf{R}_{\mathcal{T}}$ as well as the
knowledge graph $\mathcal{G}$, we aim to predict whether user $u$ will engage
with item $i$ with which the user has no interaction before. Our goal is to
learn a prediction function $\hat{y}_{ui}$ = $f$
$(u,i\mid\Theta,\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{T}},\mathcal{G})$,
where $\hat{y}_{ui}$ denotes the probability (or the rating score) that user
$u$ will engage with item $i$ and $\Theta$ denotes the model parameters of
function $f$.

###  3.3. Methodology

Figure 3. The framework of our model: KG-aware NeuCMF. It learns item
representations from both KG (left) and user-item interaction matrices
(right). Entity (item) representations learned from KG contain both domain-
specific and domain-general information by utilizing graph autoencoding
strategy, which can help assist the CDR task. Item embeddings are learned by a
neural CMF model. To ensure the two types of embeddings are highly correlated,
we maximize their MI by the neural mutual information estimator (middle).

In this subsection, we present the technical details of our proposed model,
KG-aware Neural CMF (KG-NeuCMF) that aims to improve the performance of CDR by
leveraging the KG. Fig.3 shows the overview of the proposed framework. In the
first stage, we propose to learn KG-level representations by exploiting a
multi-layer RGCN (Schlichtkrull et al., 2018) through the encode-decode
paradigm by minimizing the reconstruction loss that follows a contrastive
learning-style convention (Kipf and Welling, 2017b). This step aims to learn
item embeddings containing both domain-specific and domain-general information
from different hops of neighbors in KG. In the second-stage, we learn item and
user embeddings by borrowing ideas from the CMF framework (Singh and Gordon,
2008) and neural CF (NCF) (He et al., 2017). Instead of jointly factorizing
the two user-item interaction matrices directly as in CMF, we propose to
utilize neural networks to jointly learn the two matrices by sharing user
latent representations. Finally, item representations learned from KG and
user-item interaction matrix should be highly correlated. To quantify such
correlation, we also exploit to maximize MI (Belghazi et al., 2018) between
the two types of representations.

####  3.3.1. Entity embedding learning

To utilize the KG in our task, we first need to learn entity representations.
We do this by training a graph autoencoder model in the unsupervised fashion
and learn representations in an encode-decode paradigm (Kipf and Welling,
2017b; Schlichtkrull et al., 2018). We employ RGCN (Schlichtkrull et al.,
2018) as our encoder that learns an entity embedding by aggregating
information from its adjacent neighbors via non-linear transformation and
aggregation dependent on the connecting relation, which can be denoted as

(1)
$f_{en}(\mathbf{e}^{(l)}_{{i}},\mathbf{e}^{(l)}_{{j}})=\sigma(\mathbf{W}^{(l)}_{0}\mathbf{e}^{(l)}_{{i}}+\sum_{r\in\mathcal{R}}\sum_{{j}\in\mathcal{N}_{i}^{r}}\dfrac{1}{c_{ij}}\mathbf{W}^{(l)}_{r}\mathbf{e}^{(l)}_{{j}}),$

where $\mathbf{e}^{(l)}_{{i}},$ $\mathbf{e}^{(l)}_{{j}}$ are the hidden state
of node $i$ and node $j$ in the $l-$th layer of the encoder, $\sigma$ is an
activation function such as ReLU, $\mathbf{W}^{(l)}_{0}$,
$\mathbf{W}^{(l)}_{r}$ are (learnable parameters) relation-specific
transformation mapping matrices depending on the type of edge, $c_{ij}$ is
problem-specific normalization constant that can either be learned or chosen
in advance, and $\mathcal{N}_{i}^{r}$ denotes the set of neighbors of node $i$
under relation $r\in\mathcal{R}$. Through this operation, the local proximity
structure and related semantic information can be successfully captured and
stored in the new representation of each entity. Long-range node dependencies
can be captured by stacking multiple graph encoder layers and this mechanism
ensures that distinct domains can be connected via the information
propagation.

The decoder can be any scoring function of KG embedding methods (Wang et al.,
2017) that are used to measure the plausibility of each fact ($h,r,t$).
Following (Schlichtkrull et al., 2018), we use DisMult (Yang et al., 2015)
factorization as the scoring function, which is well known for its simplicity
and efficiency and a triple ($h,r,t$) is scored as

(2)
$f_{de}(\mathbf{e}_{{h}},\mathbf{r},\mathbf{e}_{{t}})=\mathbf{e}_{{h}}\mathbf{R}_{r}\mathbf{e}_{{t}},$

where $\mathbf{e}_{{h}},\mathbf{e}_{{t}}\in\mathbb{R}^{d}$ are encoded
features vector for entity $h$ and $t$, and each relation $\mathbf{r}$ is
associated with a diagonal matrix $\mathbf{R}_{r}\in\mathbb{R}^{d\times d}$.

We train the encoder and decoder with negative sampling. We construct an equal
number of negative samples by randomly replacing the head entity or tail
entity of each positive sample and the overall set of samples are denoted by
$\mathcal{M}$. Then we minimize the cross-entropy loss of positive and
negative node pairs

(3)
$\mathcal{L}=\sum_{(\mathbf{e}_{{h}},\mathbf{r},\mathbf{e}_{{t}},y)\in\mathcal{M}}(ylogf_{de}(\mathbf{e}_{{h}},\mathbf{r},\mathbf{e}_{{t}})))+(1-y)log(1-f_{de}(\mathbf{e}_{{h}},\mathbf{r},\mathbf{e}_{{t}})).$

####  3.3.2. NeuCMF module

Typically the user-item interaction matrices are highly sparse and it is
beneficial to learn them simultaneously (Singh and Gordon, 2008). Collective
matrix factorization (CMF) jointly factorizes two matrices by sharing the user
latent factors. Motivated by neural CF (NCF) (He et al., 2017), we propose to
utilize neural networks to jointly learn the two matrices by sharing user
latent representations as shown in Fig. 3. The predicted scores in two domains
are

(4)
$r_{ui}^{\mathcal{S}}=f_{0}(f_{u}(\mathbf{u}_{u}),f_{s}(\mathbf{i}_{i}^{\mathcal{S}}),$
(5)
$r_{uj}^{\mathcal{T}}=f_{1}(f_{u}(\mathbf{u}_{u})),f_{t}(\mathbf{i}_{j}^{\mathcal{T}})),$

where $\mathbf{u}_{u}$, $\mathbf{i}_{i}^{\mathcal{S}}$ and
$\mathbf{i}_{j}^{\mathcal{T}}$ are represented one-hot vectors of users, items
from domain $\mathcal{S}$ and domain $\mathcal{T}$ respectively. Only the
element corresponding to that index is 1 and all others are 0. $f_{u}$,
$f_{s}$ and $f_{t}$ can be multi-layer perceptron (MLP) that project sparse
representations to dense vectors. The obtained embeddings are then feed into
two separate multi-layer neural architectures to map the latent vectors to
predict scores $r_{us}^{\mathcal{S}}$, $r_{ut}^{\mathcal{T}}$ for the two
domains. Given $\mathbf{R}_{\mathcal{S}}$ and $\mathbf{R}_{\mathcal{T}}$, we
minimize the two reconstruction losses $\mathcal{L}_{\mathcal{S}}$ and
$\mathcal{L}_{\mathcal{T}}$ with the predicted scores.

The NeuCMF module connects two domains only by the common users, and fails to
capture the relations among items. The item embedding learned from KG can
capture both domain-specific and domain-general knowledge, thus will be
effective for both single-domain and cross-domain recommendation. Intuitively,
the learned item embedding from user-item interaction matrices should be
highly correlated to the KG-level embeddings. Therefore, this motivates us to
exploit to maximize MI (Belghazi et al., 2018) between the two types of
representations to guarantee their highly correlated relationship. We design
our neural mutual information estimator based on a discriminator
$\mathcal{D}(x,y)$ for their pairwise relationships, to provide probability
scores for sampled pairs. To be specific, we generate positive samples as
($\mathbf{e}_{i}$,$\mathbf{i}_{i}$) ($\mathbf{i}$ can come from domain
$\mathcal{S}$ and domain $\mathcal{T}$, half-half) and negative samples are
generated by associating sampled items with fake embeddings based on shuffling
strategy (Velickovic et al., 2019). We define the loss function as:

(6)
$\mathcal{L}_{mul}=-\frac{1}{N}(\sum_{i=1}^{N_{pos}}\mu(\mathbf{i}_{i},\mathbf{e}_{i})log\sigma(\mathbf{i}_{i},\mathbf{e}_{i})+\sum_{i=1}^{N_{neg}}\mu(\tilde{\mathbf{i}_{i}},\mathbf{e}_{i})log\sigma(\tilde{\mathbf{i}_{i}},\mathbf{e}_{i})),$

where $N=N_{pos}+N_{neg}$, $N_{pos},N_{neg}$ denotes the number of positive
and negative samples, $\mu(\cdot)$ is an indicator function,
$\sum_{i=1}^{N_{pos}}\mu(\mathbf{i}_{i},\mathbf{e}_{i})=1$ and
$\sum_{i=1}^{N_{neg}}\mu(\tilde{\mathbf{i}_{i}},\mathbf{e}_{i})=1$ corresponds
to positive and negative pair samples. We aim to minimize $\mathcal{L}_{mul}$,
which is equivalent to maximize the mutual information, to jointly preserve
the KG-level and user-item interaction information.

The final loss includes: the loss ($\mathcal{L}_{\mathcal{S}}$) of source and
loss ($\mathcal{L}_{\mathcal{T}}$) of target recommendation with the mutual
information maximization loss $\mathcal{L}_{mul}$. The objective is to
minimize the overall loss $\mathcal{L}$ as follows:

(7)
$\mathcal{L}=\mathcal{L}_{\mathcal{S}}(\Theta_{\mathcal{S}})+\mathcal{L}_{\mathcal{T}}(\Theta_{\mathcal{T}})+\mathcal{L}_{mul}(\Theta_{mul})+\lambda\lVert\Theta\rVert,$

where $\Theta$ = $\Theta_{\mathcal{S}}$ $\cup$ $\Theta_{\mathcal{T}}$ $\cup$
$\Theta_{\mathcal{L}_{mul}}$. Note that $\Theta_{\mathcal{S}}$ and
$\Theta_{\mathcal{T}}$ share user embeddings. The objective function can be
optimized by stochastic gradient descent (SGD) and its variants like adaptive
moment method (Adam) (Kingma and Ba, 2015).

##  4\. Experiment

###  4.1. Dataset

We use the Amazon Review Data (2018) (Ni et al., 2019) that is widely used for
product recommendation. It contains users’ rate (ranging from 1 to 5) for
product from various domains. We select a subset that includes two domain
pairs: movie-music(MM), movie-book(MB), which are being linked together
through a common user ID identifying the same user. We construct the knowledge
graph for each item by utilizing Freebase and take triplets that involve two-
hop neighbor entities of items into consideration. The basic statistics
details are presented in Table 1. The recommendation task can be formulate as
the regression (rating) or the binary classification (recommend or not) tasks.
Following (Ricci et al., 2011), we evaluate the recommendation performance
based MAE, F1_score (Threshold of positive rating is 4) for the regression and
classification performance, respectively.

###  4.2. Baselines

To validate the performance of the proposed model, we compare the performance
with five representative models, in which two single-domain RS models (MF,
NCF) and three CDR models (CMF, CoNet, DDTCDR) using the publicly released
implementations.

  * •

MF (Koren et al., 2009). Matrix Factorization (MF) is a classic latent factors
CF approach which learns the user and item factors via matrix factorization in
each domain separately.

  * •

NCF (He et al., 2017). Neural Collaborative Filtering (NCF) is a neural
network architecture to model latent features of users and items using CF
method. The NCF models are trained separately for each domain without
transferring any information.

  * •

CMF (Singh and Gordon, 2008). Collective Matrix Factorization (CMF) jointly
factorizes matrices of each domains. In our scenarios, The shared user factors
enable knowledge transfer between cross domains .

  * •

CoNet (Hu et al., 2018). Collaborative Cross Networks (CoNet) enables dual
knowledge transfer across domains by introducing cross connections from one
base network to another and vice versa.

  * •

DDTCDR (Li and Tuzhilin, 2020). Deep Dual Transfer Cross Domain Recommendation
(DDTCDR) learns latent orthogonal mappings across domains and provides cross
domain recommendations by leveraging user preferences from all domains.

Table 1.  Statistics of the dataset. | Domain: Music-Movie | Domain: Book-Movie
---|---|---
| Music | Movie | Book | Movie
Users | 4,196 | 4,196 | 3,977 | 3,977
Items | 7,412 | 10,919 | 11,372 | 8,118
Interactions | 21,986 | 49,027 | 22,214 | 29,245
Entities | 85,612 | 387,178 | 258,999 | 990,141
Relations | 155 | 340 | 127 | 295
Triples | 288,731 | 610,314 | 522,814 | 1,787,190

###  4.3. Implementation details

In the KG-pretrain step, we utilize a two-layer RGCN as the encoder to obtain
entity embeddings. In the NeuCMF module, we apply one-layer neural networks to
project the one-hot vectors of users, and items to low-dimensional embedding
vectors and $f_{0}$ and $f_{1}$ are two one-layer neural networks to map the
latent vectors to predict scores. Throughout the experiments, the embedding
size is tuned in the range of [8,16,32] and we use the Adam optimizer (Kingma
and Ba, 2015) with learning rate 0.001, L2 regularization 0.0001. For each
dataset, the ratio of training, evaluation, and test set is 6 : 2 : 2 (Wang et
al., 2019b). We employ the early stopping strategy based on the validation
accuracy with a window size of 10 (we will stop training if the validation
loss does not decrease for 10 consecutive epochs) and train 200 epochs at
most. We report results over 20 runs with random weight matrix initialization.
For a fair comparison, we set the same hyperparameters of the baselines as our
model.

Table 2.  Comparison of recommendation performance in Movie-Music (%). The best results are in bold and the second best ones are underlined. | Movie-Music (MM)
---|---
Methods | Movie | Music
MAE |  F1_Score |  MAE |  F1_Score
MF (Koren et al., 2009) |  20.94$\pm$2.54 |  74.97$\pm$4.50 |  23.79$\pm$1.69 |  72.57$\pm$0.75
NCF (He et al., 2017) |  19.01$\pm$0.09 |  88.93$\pm$0.05 |  15.25$\pm$3.23 |  93.05$\pm$0.43
CMF (Singh and Gordon, 2008) |  20.23$\pm$1.97 |  89.09$\pm$0.36 |  11.66$\pm$2.35 |  92.45$\pm$0.36
CoNET (Hu et al., 2018) |  18.22$\pm$0.36 |  88.68$\pm$0.70 |  13.96$\pm$0.36 |  92.05$\pm$0.48
DDTCDR (Zhu et al., 2019) |  20.69$\pm$0.35 |  74.84$\pm$1.74 |  15.82$\pm$0.75 |  89.05$\pm$2.13
Ours |  14.23$\pm$0.97 |  90.69$\pm$0.22 |  9.89$\pm$0.35 |  94.45$\pm$0.32
Improvement (%) |  21.28 % |  1.80 % |  15.18 % |  1.50%
Table 3.  Comparison of recommendation performance in Movie-Book(%). The best results are in bold and the second best ones are underlined. | Movie-Book (MB)
---|---
Methods | Movie | Book
MAE |  F1_Score |  MAE |  F1_Score
MF (Koren et al., 2009) |  24.17$\pm$1.32 |  73.64$\pm$0.74 |  23.83$\pm$1.25 |  69.01$\pm$2.74
NCF (He et al., 2017) |  18.80$\pm$0.54 |  89.08$\pm$0.07 |  18.86$\pm$0.52 |  89.35$\pm$0.06
CMF (Singh and Gordon, 2008) |  14.53$\pm$1.51 |  89.32$\pm$0.04 |  13.22$\pm$0.78 |  89.07$\pm$0.22
CoNET (Hu et al., 2018) |  17.46$\pm$0.61 |  89.59$\pm$1.45 |  17.18$\pm$0.59 |  89.22$\pm$0.77
DDTCDR (Zhu et al., 2019) |  20.17$\pm$0.56 |  82.60$\pm$2.37 |  17.15$\pm$0.54 |  90.06$\pm$0.39
Ours |  13.17$\pm$0.16 |  90.60$\pm$0.37 |  13.01$\pm$0.14 |  90.80$\pm$0.22
Improvement (%) |  9.36 % |  1.12 % |  1.58 % |  0.57%
Figure 4. Different ways to incorporate KG information for CDR.

###  4.4. Overall Performance of CDR

We have conducted experiments on two cross domain tasks, movie-music (MM) and
movie-book (MB), and the corresponding results of our model and baselines are
shown in Table 2 and Table 3. We can see that our proposed model can
consistently obtain the best performance across movie-music and movie-book
recommendations in terms of MAE and F1_score. In particular, our model
improves over the strongest baselines $w.r.t.$ MAE by 21%, 15.18% in movie,
music (Table 2) respectively, which justifies the effectiveness of our method
in integrating items’ KG information. If we compare between these two tasks,
MM and MB, the improvement on music in MM is more remarkable compared to the
performance in MB. Possible reasons are 1) the data is more sparse in the
user-music interaction matrix, so leveraging KG information can greatly
relieve the sparsity problem (we have verified this in the later experiments:
Comparisions for cold-start item scenarios); 2) the extracted KG contains much
useful information, especially for two closely related domains (movie and
music both belong to multi-media datasets). Besides, CDR models
(CMF,CoNet,DDTCDR) achieve better performance than SDR models (MF, NCF),
indicating that utilizing extra information from other resources benefits the
performance of recommendation.

###  4.5. Different ways to incorporate KG

We explore different ways to combine item embeddings learned from KG and user-
item interaction matrices. NMF_KG takes KG-level embeddings as input, then
incorporates them with item embeddings learned from user-item interaction
matrices via an aggregation method, e.g., concatenation. NCMF_KG_T tries to
refine item embeddings learned from KG with a one-layer MLP and concatenates
with embeddings learned from the user-item interaction matrix. NCMF_KG_mul
maximizes MI between the two types of representations to guarantee the highly
correlated relationship. The results are shown in Fig. 4. Generally, refining
the learned KG-level embeddings gets better performance than direct
utilization. This is because in real-world KGs (e.g., Freebase) some noises
are inevitably introduced in the process of automatically constructing large-
scale KGs due to limited labour supervision (Xie et al., 2018; Jia et al.,
2019). NCMF_KG_mul gets the best performance. The possible reason is that item
embeddings jointly learn from the user-item rating matrix and entity
embeddings from KG, which contain both domain-general and domain-specific
knowledge and the neural mutual information estimator can ensure their
correlation. Such design is more suitable for the cross-domain recommendation
task.

###  4.6. Comparisons for cold-start item scenarios

The KG a natural bridge for items from different domains, which can further
alleviate the item cold-start problem in RS.

To validate this, we compare our methods with NCF, CMF under the code-start
scenario. We set up the cold-start environment by sampling a subset of items
for testing which are unseen in the training data. Results for cold-start
items on movie-music datasets are shown in Fig. 5. NCF (the SDR model) is
greatly influenced and gets the poorest performance, especially there are a
large proportion new items. CMF (the CDR model) can leverage information from
two domains, thus it can alleviate the cold-start problem in some extent. Our
model goes further to learn representations for cold items from the KG,
offering additional information beyond user-item interaction matrices.

(a) Movie (in MM).

(b) Music (in MM).

Figure 5. Comparison of different models in cold-start items scenarios.

##  5\. Conclusion

In this paper, we constructed a new dataset AmazonKG4CDR, the first in the
filed linking KG information for cross-domain recommendation. Moreover, we
proposed a KG-aware NeuCMF model to learn domain-specific and domain-general
knowledge using graph autoencoding strategy to capture both adjacent and
higher-order neighborhood information from KG. Our model unified item
embeddings learned from user-item interaction matrices and KG with a neural
collaborative filtering framework under a mutual information-based neural
estimator. Through extensive experiments on real-world datasets, we
demonstrated that KG-aware NeuCMF has achieved substantial gains over state-
of-the-art baselines. For future work, we will explore the explainability of
cross-domain recommendation.

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# Rationality of even-dimensional intersections of two real quadrics

Brendan Hassett  Department of Mathematics
Brown University
Box 1917 151 Thayer Street Providence, RI 02912
USA  brendan<EMAIL_ADDRESS>,  János Kollár  Department of Mathematics
Princeton University
Fine Hall, Washington Road
Princeton NJ 08544-1000
USA<EMAIL_ADDRESS>and  Yuri Tschinkel  Courant Institute
New York University
New York, NY 10012
USA  Simons Foundation
160 Fifth Avenue
New York, NY 10010
USA<EMAIL_ADDRESS>
###### Abstract.

We study rationality constructions for smooth complete intersections of two
quadrics over nonclosed fields. Over the real numbers, we establish a
criterion for rationality in dimension four.

##  1\. Introduction

Consider a geometrically rational variety $X$, smooth and projective over a
field $k$. Is $X$ rational over $k$? A necessary condition is that
$X(k)\neq\emptyset$, which is sufficient in dimension one, as well as for
quadric hypersurfaces and Brauer-Severi varieties of arbitrary dimension. When
the dimension of $X$ is at most two, rationality over $k$ was settled by work
of Enriques, Manin, and Iskovksikh [Isk79]. Rationality is encoded in the
Galois action on the geometric Néron-Severi group – varieties with rational
points that are ‘minimal’ in the sense of birational geometry need not be
rational. In dimension three, recent work [HT19a, HT19c, KP20a, BW20, BW19]
has clarified the criteria for rationality: one also needs to take into
account principal homogeneous spaces over the intermediate Jacobian,
reflecting which curve classes are realized over the ground field. The case of
complete intersections of two quadrics was an important first step in
understanding the overall structure [HT19b]; rationality in dimension three is
equivalent to the existence of a line over $k$ [HT19a, BW19, KP20a].

These developments stimulate investigations in higher dimensions [KP20b]; the
examples considered are rational provided there are rational points. In this
paper, we focus on the case of four-dimensional complete intersections of two
quadrics, especially over the real numbers $\mathbb{R}$. Here we exhibit
rational examples without lines and explore further rationality constructions.

######  Theorem 1.1.

A smooth complete intersection of two quadrics $X\subset\mathbb{P}^{6}$ over
$\mathbb{R}$ is rational if and only if $X(\mathbb{R})$ is nonempty and
connected.

In general, a projective variety $X$ that is rational over $\mathbb{R}$ has
connected nonempty real locus $X(\mathbb{R})$. The point of Theorem 1.1 is
that this necessary condition is also sufficient.

######  Corollary 1.2.

A smooth complete intersection of two quadrics $X\subset\mathbb{P}^{6}$ is
rational over $\mathbb{R}$ if and only if there exists a unirational
parametrization $\mathbb{P}^{4}\dashrightarrow X$, defined over $\mathbb{R}$,
of odd degree.

Indeed, odd degree rational maps are surjective on real points, which
guarantees that $X(\mathbb{R})$ is connected. Smooth complete intersections of
two quadrics, of dimension at least two, are unirational provided they have a
rational point; see Section 3.1 for references and discussion.

We also characterize rationality in dimension six, with the exception of one
isotopy class that remains open (see Section 6.2).

Here is the roadmap of this paper. In Section 2 we recall basic facts about
quadrics in even-dimensional projective spaces and their intersections. All
interesting cohomology is spanned by the classes of projective subspaces in
$X$ of half-dimension, and the Galois group acts on these classes via
symmetries of the primitive part of this cohomology, a lattice for the root
system $D_{2n+3}$. In Section 3 we present several rationality constructions.
The isotopy classification, using Krasnov’s invariants, is presented in
Section 4; we draw connections with the Weyl group actions. In Section 5 we
focus attention on cases where rationality is not obvious, e.g., due to the
presence of a line. In Section 6 we prove Theorem 1.1 and discuss the
applicability and limitations of our constructions in dimensions four and six.
We speculate on possible extensions to more general fields in Section 7.

Acknowledgments: The first author was partially supported by NSF grant 1701659
and Simons Foundation Award 546235, the second author by the NSF grant
DMS-1901855, and the third author by NSF grant 2000099.

##  2\. Geometric background

###  2.1. Roots and weights

Let $D_{2n+3}$ be the root lattice of the corresponding Dynkin diagram,
expressed in the standard Euclidean lattice

$\left<L_{1},\ldots,L_{2n+3}\right>,\quad\quad L_{i}\cdot L_{j}=\delta_{ij}$

as the lattice generated by simple roots

$\displaystyle R_{1}=L_{1}-L_{2},R_{2}=L_{2}-L_{3},$
$\displaystyle\ldots,R_{2n+1}=L_{2n+1}-L_{2n+2},$ $\displaystyle
R_{2n+2}=L_{2n+2}-L_{2n+3},\quad$ $\displaystyle R_{2n+3}=L_{2n+2}+L_{2n+3}.$

Its discriminant group is cyclic of order four, generated by

$\tfrac{1}{4}(2(R_{1}+2R_{2}+\cdots+(2n+1)R_{2n+1})+(2n+1)R_{2n+2}+(2n+3)R_{2n+3}).$

Multiplication by $-1$ acts on the discriminant via $\pm 1$. The outer
automorphisms of $D_{2n+3}$ also act via automorphisms of $D_{2n+3}$ acting on
the discriminant via $\pm 1$, e.g., exchanging $R_{2n+2}$ and $R_{2n+3}$ and
keeping the other roots fixed.

The Weyl group $W(D_{2n+3})$ acts in the basis $\\{L_{i}\\}$ via signed
permutations with an even number of $-1$ entries. The outer automorphisms act
via signed permutations with no constraints on the choice of signs, e.g.,

$L_{i}\mapsto L_{i},\quad i=1,\ldots,2n+2,\quad\quad L_{2n+3}\mapsto-
L_{2n+3}.$

The odd and even half-spin representations have weights indexed by subsets
$I\subset\\{1,2,\ldots,2n+3\\}$, with $|I|$ odd or even, written

$w_{I}=\tfrac{1}{2}(\sum_{i\in I}L_{i}-\sum_{j\in I^{c}}L_{j}).$

The odd and even weights are exchanged by outer automorphisms.

###  2.2. Planes

In this section, we assume that the ground field is algebraically closed of
characteristic zero.

Let $X\subset\mathbb{P}^{2n+2}$ be a smooth complete intersection of two
quadrics. We will identify subvarieties in $X$ with their classes in the
cohomology of $X$ when no confusion may arise.

Let $h$ denote the hyperplane section and consider the primitive cohomology of
$X$ under the intersection pairing. Reid [Rei72, 3.14] shows that

$(h^{n})^{\perp}\simeq(-1)^{n}D_{2n+3}.$

In other words, the primitive sublattice of $H^{2n}(X,\mathbb{Z}(n))$ – the
Tate twist of singular cohomology for the underlying complex variety – may be
identified with the root lattice. This is the target of the cycle class map

$\operatorname{CH}^{n}(X)\rightarrow H^{2n}(X,\mathbb{Z}(n))$

so the sign convention is natural.

######  Remark 2.1 (Caveat on signs).

When $X$ is defined over $\mathbb{R}$, codimension-$n$ subvarieties $Z\subset
X$ defined over $\mathbb{R}$ yield classes in $H^{2n}(X,\mathbb{Z}(n))$ that
are invariant under complex conjugation. However, the corresponding classes in
$H^{2n}(X,\mathbb{Z})$ are multiplied by $(-1)^{n}$. When we mention invariant
classes, it is with respect to the former action.

Given a plane $P\simeq\mathbb{P}^{n}\subset X$, we have

$(P\cdot P)_{X}=c_{n}$

where [Rei72, 3.11]

$c_{0}=1,c_{1}=-1,c_{2}=2,c_{3}=-2,\ldots,c_{n}=(-1)^{n}(\lfloor\tfrac{n}{2}\rfloor+1).$

The projection of $P$ into rational primitive cohomology takes the form

$P-\tfrac{1}{4}h^{n}$

which has self-intersection $c_{n}-1/4$. The corresponding element $w_{P}\in
D_{2n+3}$ has

$w_{P}\cdot w_{P}=(2n+3)/4.$

By [Rei72, Cor. 3.9], we obtain bijections

$\\{w_{P}\\}_{P\simeq\mathbb{P}^{n}\subset X}=\\{w_{I}\\}_{|I|\text{ has fixed
parity }}.$

Note that the residual intersections to $\mathbb{P}^{n}\subset X$

$X\cap\mathbb{P}^{n+2}=\mathbb{P}^{n}\cup S$

give cubic scrolls $S\subset X$; these realize the weights of opposite parity.

By [Rei72, Th. 3.8], two planes $P_{1}$ and $P_{2}$ are disjoint if and only
if

$w_{P_{1}}\cdot w_{P_{2}}=(-1)^{n+1}/4.$

For example, if $n=1$ and $w_{P_{1}}$ is identified with
$(L_{1}-L_{2}-L_{3}-L_{4}-L_{5})/2$ then the relevant weights are

$(L_{1}+L_{2}+L_{3}-L_{4}-L_{5})/2,\quad\ldots,\quad(L_{1}-L_{2}-L_{3}+L_{4}+L_{5})/2$

and

$(-L_{1}+L_{2}+L_{3}+L_{4}-L_{5})/2,\quad\ldots,\quad(-L_{1}-L_{2}+L_{3}+L_{4}+L_{5})/2,$

a total of $10=\binom{5}{2}$ such lines. When $n=2$ and $w_{P_{1}}$ is
identified with $(L_{1}-L_{2}-\cdots-L_{7})/2$ then the relevant weights are

$(L_{1}+L_{2}+L_{3}+L_{4}+L_{5}-L_{6}-L_{7})/2,\ldots,(L_{1}-L_{2}-L_{3}+L_{4}+L_{5}+L_{6}+L_{7})/2$

and

$(-L_{1}+L_{2}+L_{3}+L_{4}-L_{5}-L_{6}-L_{7})/2,\ldots,(-L_{1}-L_{2}-L_{3}-L_{4}+L_{5}+L_{6}+L_{7})/2,$

a total of $\binom{6}{4}+\binom{6}{3}=35=\binom{7}{3}$ planes. The planes
$P_{1}$ and $P_{2}$ meet at a point if and only if

$w_{P_{1}}\cdot w_{P_{2}}=(-1)^{n}\tfrac{3}{4}.$

If they meet along an $r$-plane then an excess intersection computation gives
[Rei72, 3.10]

$P_{1}\cdot P_{2}=(-1)^{r}(\lfloor\tfrac{r}{2}\rfloor)+1)$

and

$w_{P_{1}}\cdot
w_{P_{2}}=(-1)^{r+n}(\lfloor\tfrac{r}{2}\rfloor+1)-(-1)^{n}\tfrac{1}{4}.$

In particular, they meet along an $(n-1)$-plane when

$w_{P_{1}}\cdot
w_{P_{2}}=-(\lfloor\tfrac{n-1}{2}\rfloor+1)-(-1)^{n}\tfrac{1}{4};$

for a fixed $w_{P_{1}}$ there are $2n+3$ planes $P_{2}$ meeting $P_{1}$ in
this way. For example, if $n=1$ and
$w_{P_{1}}=(L_{1}-L_{2}-L_{3}-L_{4}-L_{5})/2$ then the possibilities for
$w_{P_{2}}$ are

$\displaystyle(L_{1}+L_{2}+L_{3}+L_{4}+L_{5})/2,\quad(-L_{1}-L_{2}+L_{3}+L_{4}+L_{5})/2,$
$\displaystyle(-L_{1}+L_{2}-L_{3}+L_{4}+L_{5})/2,\quad(-L_{1}+L_{2}+L_{3}-L_{4}+L_{5})/2,$
$\displaystyle(-L_{1}+L_{2}+L_{3}+L_{4}-L_{5})/2.$

###  2.3. Quadrics

We retain the notation of Section 2.2.

Our next task is to analyze quadric $n$-folds $Q\subset X$, i.e., $Q$ a
degree-two hypersurface in $\mathbb{P}^{n+1}$. Let
$\\{\mathcal{Q}_{t}\\},t\in\mathbb{P}^{1}$ denote the pencil of quadric
hypersurfaces cutting out $X$. The degeneracy locus

$D:=\\{t\in\mathbb{P}^{1}:\mathcal{Q}_{t}\text{ singular }\\}$

consists of $2n+3$ points; since $X$ is smooth, each has rank $2n+2$. The
Hilbert scheme of quadric $n$-folds $Q\subset X$ is isomorphic to the relative
Fano variety of $(n+1)$-planes

$\mathcal{F}(\mathcal{Q}/\mathbb{P}^{1})=\\{\Pi\simeq\mathbb{P}^{n+1}\subset\mathcal{Q}_{t}\text{
for some }t\in\mathbb{P}^{1}\\},$

which consists of $2(2n+3)$ copies of the connected isotropic Grassmannian
$\operatorname{OGr}(n+1,2n+2)$. Given a quadric $Q$, its projection to
rational primitive cohomology

$Q-\tfrac{1}{2}h^{n}$

corresponds to an element

$w_{Q}\in D_{2n+3},\quad w_{Q}\in\\{\pm L_{1},\ldots,\pm L_{2n+3}\\}.$

In particular, we have

$Q\cdot Q=\begin{cases}2&\text{ if $n$ even}\\\ 0&\text{ if $n$
odd.}\end{cases}$

Residuation in a complete intersection of linear forms

$Q\cup Q^{\prime}=X\cap h^{n}$

reverses signs, i.e., $w_{Q}=-w_{Q^{\prime}}$. On the other hand, if $Q_{1}$
and $Q_{2}$ are not residual then

(2.1) $Q_{1}\cdot Q_{2}=1.$

We summarize this as follows:

######  Proposition 2.2.

The signed permutation representation of $W(D_{2n+3})$ is realized via the
action on classes $[Q]$, where $Q\subset X$ is a quadric $n$-fold.

Note that there are

$2^{2n+1}(2n+3)$

reducible quadrics – unions of two $n$-planes meeting in an $(n-1)$-plane –
with $2^{2n}$ reducible quadrics in each copy of the isotropic Grassmannian.

##  3\. Rationality constructions

We now work over an arbitrary field $k$ of characteristic zero.

###  3.1. General considerations

Let $X\subset\mathbb{P}^{d+2}$ denote a smooth complete intersection of two
quadrics of dimension at least two. Recall the following:

  * •

If $X(k)\neq\emptyset$ then $X$ is unirational over $k$ and has Zariski dense
rational points [CTSSD87, Rem. 3.28.3].

  * •

If there is a line $\ell\subset X$ defined over $k$ then projection induces a
birational map
$\pi_{\ell}:X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\mathbb{P}^{d}$.

For reference, we recall Amer’s theorem [Lee07, Th. 2.2]:

######  Theorem 3.1.

Let $k$ be a field of characteristic not two, $F_{1}$ and $F_{2}$ quadrics
over $k$, and $\mathcal{Q}_{t}=\\{F_{1}+tF_{2}\\}$ the associated pencil of
quadrics over $k(t)$. Then $X=\\{F_{1}=F_{2}=0\\}$ has an $r$-dimensional
isotropic subspace over $k$ if and only if $\mathcal{Q}_{t}$ has an
$r$-dimensional isotropic subspace over $k(t)$.

We apply this for $k=\mathbb{R}$, where $X\subset\mathbb{P}^{d+2}$ is a smooth
complete intersection of two quadrics and
$\mathcal{Q}\rightarrow\mathbb{P}^{1}$ is the associated pencil.

Recall Springer’s Theorem: A quadric hypersurface $\mathcal{Q}$ over a field
$L$ has a rational point if it admits a rational point over some odd-degree
extension of $L$. Applying this to the pencil
$\mathcal{Q}\rightarrow\mathbb{P}^{1}$ associated with a complete intersection
of two quadrics, with Amer’s Theorem, yields:

######  Proposition 3.2.

If $d\geq 1$ and $X$ contains a subvariety of odd degree over $k$ then
$X(k)\neq\emptyset$.

We can prove a bit more:

######  Proposition 3.3.

If $d\geq 2$ and $X$ has a curve of odd degree defined over $k$ then $X$ is
rational over $k$.

###### Proof.

Recall that double projection from a sufficiently general rational point $x\in
X(k)$ yields a diagram

$X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}Y\rightarrow\mathbb{P}^{1}$

where $Y$ is a quadric bundle of relative dimension $d-1$. A curve $C\subset
X$ of odd degree yields an multisection of this bundle of odd degree. Thus
$Y\rightarrow\mathbb{P}^{1}$ has a section by Springer’s Theorem and its
generic fiber $Y_{t}$ is rational over $k(\mathbb{P}^{1})$. It follows that
$X$ is rational over the ground field. ∎

######  Remark 3.4.

The pencil defining $X$ gives a quadric bundle

$\mathcal{Q}\rightarrow\mathbb{P}^{1}$

of relative dimension $d+1$. We apply Witt’s decomposition theorem to
$[\mathcal{Q}_{t}]$ and $[Y_{t}]$, understood as quadratic forms over
$k(\mathbb{P}^{1})=k(t)$, to obtain

$[\mathcal{Q}_{t}]=[Y_{t}]\oplus\left(\begin{matrix}0&1\\\
1&0\end{matrix}\right).$

Thus a section of $Y\rightarrow\mathbb{P}^{1}$ yields an isotropic line of
$\mathcal{Q}\rightarrow\mathbb{P}^{1}$, and Theorem 3.1 implies that $X$
contains a line defined over $k$.

######  Corollary 3.5.

(see appendix by Colliot-Thélène [HT19b, Th. A5]) Let
$X\subset\mathbb{P}^{d+2}$ denote a smooth complete intersection of two
quadrics of dimension at least two. Suppose there exists an irreducible
positive-dimensional subvariety $W\subset X$ of odd degree, defined over $k$.
Then $X$ is rational over $k$.

Given these results, we focus on proving rationality in cases where $X$ does
not contain lines or positive-dimensional subvarieties of odd degree.

###  3.2. Rationality using half-dimensional subvarieties

We now turn to even-dimensional intersections of two quadrics

$X\subset\mathbb{P}^{2n+2},\quad n\geq 1.$

Throughout, we assume that $X(k)\neq\emptyset,$ and thus $X$ is
$k$-unirational and $k$-rational points are Zariski dense.

Construction I: Suppose that

  * •

$X$ has a pair of conjugate disjoint $n$-planes $P,\bar{P}$, defined over a
quadratic extension $K$ of $k$.

Projecting from a general $x\in X(k)$ gives a birational map

$X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X^{\prime}\subset\mathbb{P}^{2n+1},$

where $X^{\prime}$ is a cubic hypersurface.

Since $X(k)\subset X$ is Zariski dense, we may assume that the images of $P$
and $\bar{P}$ in $X^{\prime}$ remain disjoint. The ‘third point’ construction
gives a birational map

$\mathbf{R}_{K/k}(P)\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X^{\prime},$

where the source variety is the restriction of scalars. We conclude that $X$
is rational over $k$. This construction appears in [CTSSD87, Th. 2.4].

Construction II: Suppose that

  * •

$X$ has a pair of conjugate disjoint quadric $n$-folds $Q,\bar{Q}$, defined
over a quadratic extension $K$ of $k$, and meeting transversally at one point.

Projecting from $x\in Q\cap\bar{Q}$, which is a $k$-rational point $X$, gives
a birational map

$X\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X^{\prime}\subset\mathbb{P}^{2n+1},$

where $X^{\prime}$ is a cubic hypersurface.

The proper transforms $Q^{\prime},\bar{Q}^{\prime}\subset X^{\prime}$ are
disjoint unless there exists a line

$x\in\ell\subset\mathbb{P}^{2n+2}$

with

$\\{x\\}\subsetneq\ell\cap Q,\ell\cap\bar{Q}$

as schemes. We may assume that $\ell\not\subset X$ as we already know $X$ is
rational in this case. Thus the only possibility is

$\ell\cap Q=\ell\cap\bar{Q}$

as length-two subschemes, which is precluded by the intersection assumption.

Repeating the argument for Construction I thus gives rationality.

Construction III: Suppose that

  * •

$X$ contains a quadric $Q$ of dimension $n$, defined over $k$.

Projection gives a fibration

$q:\operatorname{Bl}_{Q}(X)\rightarrow\mathbb{P}^{n}$

with fibers quadrics of dimension $n$. Now suppose that $X$ contains a second
$n$-fold $T$ with the property that

$\deg(T)-T\cdot Q$

is odd, i.e., a multisection for $q$ of odd degree. It follows that the
generic fiber of $q$ is rational and thus $Y$ is rational over $k$.

When $\dim(X)=4$ a number of $T$ might work, e.g.,

  * •

a plane disjoint from $Q$,

  * •

a second quadric meeting $Q$ in one point,

  * •

a quartic or a sextic del Pezzo surface meeting $Q$ in one or three points,

  * •

a degree 8 K3 surface meeting $Q$ in one, three, five, or seven points.

Construction IV: Suppose that

  * •

$\dim(X)=4$ and $X$ contains a quartic scroll $T$, defined over $k$.

Geometrically, $T$ is the image of the ruled surface

${\mathbb{F}}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}\hookrightarrow\mathbb{P}^{5}$

under the linear series of bidegree $(1,2)$. Over $\mathbb{R}$ we are
interested in cases where $T=\mathbb{P}^{1}\times C$ with $C$ a nonsplit
conic. We do not want to force $X$ to have lines! (Note that quartic scrolls
geometrically isomorphic to ${\mathbb{F}}_{2}$ contain lines defined over the
ground field and thus are not useful for our purposes.)

On projecting from a point $x\in X(k)$ we get a cubic fourfold

$X^{\prime}\subset\mathbb{P}^{5},$

containing a quartic scroll. The Beauville-Donagi construction [BD85] –
concretely, take the image under the linear system of quadrics vanishing along
$T$ – shows that $X^{\prime}$ is birational to a quadric hypersurface thus
rational over $k$.

Recall that an $n$-dimensional smooth variety $W\subset\mathbb{P}^{2n+1}$ is
said to have one apparent double point if a generic point is contained in a
unique secant to $W$.

Construction V: Suppose that $X$ contains a variety $W$ defined over $k$ of
dimension $n\geq 2$ such that:

  * •

$W$ spans a $\mathbb{P}^{2n+1}$ and has one apparent double point; or

  * •

$W$ has a rational point $w$ such that projecting from $w$ maps $W$
birationally to a variety with one apparent double point.

Then $X$ is rational over $k$.

As before, one projects from a rational point to a get cubic hypersurface
$X^{\prime}\subset\mathbb{P}^{2n+1}$. Cubic hypersurfaces containing varieties
$W$ with one apparent double point are rational [Rus00, Prop. 9]. Indeed,
intersecting secant lines of $W$ with $X^{\prime}$ yields

$\operatorname{Sym}^{2}(W)\dashrightarrow X^{\prime},$

which is birational when each point lies on a unique secant to $W$.

Quartic scrolls in $\mathbb{P}^{5}$ have one apparent double point so
Construction IV is a special case of Construction V.

##  4\. Isotopy classification

We review the classification of smooth complete intersections of two quadrics
$X\subset\mathbb{P}^{2n+2}$ over $\mathbb{R}$, following [Kra18].

Express $X=\\{F_{1}=F_{2}=0\\}$ where $F_{1}$ and $F_{2}$ are real quadratic
forms. We continue to use $D$ for the degeneracy locus of the associated
pencil $\mathcal{Q}_{t}=\\{t_{1}F_{1}+t_{2}F_{2}=0\\}$. Let
$r=|D(\mathbb{R})|$ which is odd with $r\leq 2n+3$. Consider the signatures
$(I^{+},I^{-})$ of the forms

$s_{1}F_{1}+s_{2}F_{2},\quad(s_{1},s_{2})\in\mathbb{S}^{1}=\\{(s_{1},s_{2})\in\mathbb{R}^{2}:s_{1}^{2}+s_{2}^{2}=1\\}.$

Record these at the $2r$ points lying over $D$, in order as we trace the
circle counterclockwise. We label each of these points with $\pm$ depending on
whether the positive part $I^{+}$ of the signature increases or decreases as
we cross the point. Each point of $D(\mathbb{R})$ yields a pair of antipodal
points on $\mathbb{S}^{1}$ labelled with opposite signs. For example, for
$n=0$ and $r=3$ admissible sequences of signatures and $\pm$’s include

$(0,2)(1,1)(2,0)(2,0)(1,1)(0,2)\quad(+,+,+,-,-,-).$

and

$(1,1)(1,1)(1,1)(1,1)(1,1)(1,1)\quad(+,-,+,-,+,-).$

Suppose the sequence of $\pm$’s has maximal unbroken chains of $+$’s of
lengths $r_{1},r_{2},\ldots,r_{2s+1}$ where

$r=r_{1}+r_{2}+\cdots+r_{2s+1}.$

The number of terms is odd because antipodal points have opposite signs. In
the examples above we have $3=3$ and $3=1+1+1$. Our invariant is the sequence
$(r_{1},\ldots,r_{2s+1})$ up to cyclic permutations and reversals – a complete
isotopy invariant of $X$ over $\mathbb{R}$ [Kra18].

We derive a sequence of $\pm 1$’s of length $k$ from this invariant as
follows: For each point of $D(\mathbb{R})$, record the sign of the
discriminant of the associated rank-$(2n+2)$ quadratic form, determined by the
parity of $(I^{+}-I^{-})/2$. In the examples above, we obtain $(-1,+1,-1)$ and
$(+1,+1,+1)$. The number of $-1$’s is always even.

The analysis in Section 2.3 shows that complex conjugation acts on
$H^{2n}(X,\mathbb{Z}(n))$ in the basis $\\{L_{1},\ldots,L_{2n+3}\\}$ as a
signed permutation of order two. This is a direct sum of blocks

$(+1),(-1),\pm\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right).$

Actually, we may assume the sign is positive in the third case after
conjugating by

$\left(\begin{matrix}0&-1\\\ 1&0\end{matrix}\right).$

Suppose there are $a$ blocks $(+1)$, $2b$ blocks $(-1)$, and $c$ blocks of the
third kind, with $a+2b+2c=2n+3$. These correspond to the conjugacy classes of
involutions $\iota\in W(D_{2n+3})$ [Kot00, §3.2,3.3]. We have $r=a+2b$,
reflecting the number of points of $D(\mathbb{R})$ with positive and negative
discriminants respectively, and $2c=2n+3-r$, reflecting the number of complex-
conjugate pairs in $D(\mathbb{C})\setminus D(\mathbb{R})$.

The passage from isotopy classes to conjugacy classes in $W(D_{2n+3})$ results
in a loss of information. We give an example for real quartic del Pezzo
surfaces $X\subset\mathbb{P}^{4}$.

######  Example 4.1.

The isotopy class $(5)$ has singular members with signatures

$(0,4)(1,3)(2,2)(3,1)(4,0)(4,0)(3,1)(2,2)(1,3)(0,4)$

with involution given the diagonal $5\times 5$ matrix

$\operatorname{diag}(1,-1,{\bf{1}},-1,1),$

where the bolded $\bf{1}$ corresponds to a degenerate fiber $\mathcal{Q}_{t}$
whose rulings sweep out quadric curves (conics) on $X$ defined over
$\mathbb{R}$. (There is only one pair of such conics.) Here
$X(\mathbb{R})=\emptyset$ as it is contained in an anisotropic quadric
threefold.

The isotopy class $(2,2,1)$ has singular members with signatures

$(1,3)(2,2)(2,2)(2,2)(3,1)(3,1)(2,2)(2,2)(2,2)(1,3)$

with involution

$\operatorname{diag}(-1,{\bf{1}},{\bf{1}},{\bf{1}},-1).$

This has the same Galois action but contains three pairs of conics defined
over $\mathbb{R}$, represented by the bolded $\bf{1}$’s.

These are distinguished cohomologically by the arrow

$\mathbb{Z}^{3}\simeq H^{0}(G,\operatorname{Pic}(X_{\mathbb{C}}))\rightarrow
H^{2}(G,\Gamma(\mathcal{O}^{*}_{X_{\mathbb{C}}}))=\operatorname{Br}(\mathbb{R})\simeq\mathbb{Z}/2\mathbb{Z}$

of the Hochschild-Serre spectral sequence.

######  Proposition 4.2.

Fix a conjugacy class $[\iota=\iota_{abc}]$ of involutions in $W(D_{2n+3})$.
Consider the isotopy classes of $X(\mathbb{R})\subset\mathbb{P}^{2n+2}$ such
that complex conjugation acts by $\iota$. The possible isotopy classes
correspond to shuffles of

$(\underbrace{1,\ldots,1}_{a\text{ times}},\underbrace{-1,\ldots,-1}_{2b\text{
times}})$

up to cyclic permutations and reversals.

###### Proof.

Observe first that points in $D(\mathbb{C})\setminus D(\mathbb{R})$ are
irrelevant to the Krasnov invariant, assuming the dimension is given. So it
makes sense they are not relevant in the enumeration.

We have already seen that sequences of the prescribed form arise from each
isotopy class; we present the reverse construction.

Choose such a sequence, e.g.

$(1,1,1,-1,1,-1,-1,1,-1).$

The key observation is that local maxima and minima of $I^{+}-I^{-}$ – which
necessarily occur at smooth points – arise precisely between points of the
degeneracy locus where signs do not change. We indicate smooth fibers
achieving maxima/minima with $\mid$, e.g.,

$1\mid 1\mid 1,-1,1,-1\mid-1,1,-1$

or equivalently

$\mid 1\mid 1,-1,1,-1\mid-1,1,-1,1\mid$

after cyclic permutation.

Lifting to the double cover $\mathbb{S}^{1}$ entails concatenating two such
expressions:

$\mid 1\mid 1,-1,1,-1\mid-1,1,-1,1\mid 1\mid 1,-1,1,-1\mid-1,1,-1,1\mid$

From this, we read off the points of $D(\mathbb{R})$ on which $I^{+}$
increases and decreases

$+----++++-++++----$

which determines the Krasnov invariant – $(1,4,4)$ in this example. ∎

##  5\. Applying quadratic forms

###  5.1. Quadric fibrations over real curves

Let $C$ be a smooth projective geometrically connected curve over $\mathbb{R}$
with function field $K=\mathbb{R}(C)$. Let $Q\subset\mathbb{P}^{d+1}$ be a
smooth (rank $d+2$) quadric hypersurface over $K$ and $F_{i}(Q)$ the
$i$-dimensional isotropic subspaces, so that $F_{0}(Q)=Q$ and $F_{m}(Q)$ is
empty when $2m>d$. If $d=2m$ then $F_{m}(Q)$ has two geometrically connected
components; otherwise it is connected.

Suppose that $\pi:\mathcal{Q}\rightarrow C$ is a regular projective model of
$Q$, such that the fibers are all quadric hypersurfaces of rank at least
$d+1$. The locus $D\subset C$ corresponding to fibers of rank $d+1$ is called
the degeneracy locus.

Fundamental results of Witt – see [CT96, §2] and [Sch96] for a modern
formulation in terms of local-global principles – assert that:

  * •

if $d>0$ then $Q(K)\neq\emptyset$ if $\mathcal{Q}_{c}=\pi^{-1}(c)$ has a
smooth real point for each $c\in C(\mathbb{R})$;

  * •

if $d>2$ then $F_{1}(Q)(K)\neq\emptyset$ if $F_{1}(\mathcal{Q}_{c})$ has a
smooth real point for each $c\in C(\mathbb{R})$.

We can translate these into conditions on the signatures of the smooth fibers

  * •

if $d>0$ then $Q(K)\neq\emptyset$ if $\mathcal{Q}_{c}$ is not definite for any
$c\in(C\setminus D)(\mathbb{R})$;

  * •

if $d>2$ then $F_{1}(Q)(K)\neq\emptyset$ if $\mathcal{Q}_{c}$ does not have
signatures $(d+2,0),(d+1,1),(1,d+1)$ or $(0,d+2)$ for any $c\in(C\setminus
D)(\mathbb{R})$.

In other words, we have points and lines over $K$ if the fibers permit them.

This reflects a general principle: Suppose $\mathcal{X}$ is regular and has a
flat proper morphism $\varpi:\mathcal{X}\rightarrow C$ to a curve $C$, all
defined over $\mathbb{R}$. The local-global and reciprocity obstructions to
sections of $\varpi$ are reflected in the absence of continuous sections
$C(\mathbb{R})\rightarrow\mathcal{X}(\mathbb{R})$ for induced map of the
underlying real manifolds [Duc98].

###  5.2. Implications of Amer’s Theorem

Let $X\subset\mathbb{P}^{d+2}$ be a smooth complete intersection of two
quadrics over $\mathbb{R}$ and $\mathcal{Q}\rightarrow\mathbb{P}^{1}$ the
associated pencil of quadrics.

The results of Section 5.1 imply that $\mathcal{Q}\rightarrow\mathbb{P}^{1}$
has a section unless the Krasnov invariant is $(d+3)$; the variety of lines
$F_{1}(\mathcal{Q}/\mathbb{P}^{1})\rightarrow\mathbb{P}^{1}$ has a section
unless the Krasnov invariant is

$(d+3),(d+2-e,e,1)\text{ with }1\leq e\leq\tfrac{d+2}{2},\quad(d+1).$

Thus the Krasnov invariant determines which dimensional linear subspaces and
quadrics appear on $X$:

######  Proposition 5.1.

Let $X\subset\mathbb{P}^{d+2}$ be a smooth complete intersection of two
quadrics defined over $\mathbb{R}$. The only isotopy classes of $X$ that fail
to contain a line are:

  * •

$(d+3)$ where $X(\mathbb{R})=\emptyset$;

  * •

$(d+2-e,e,1)$ with $1\leq e\leq\tfrac{d+2}{2}$;

  * •

$(d+1)$.

The case $(d+1,1,1)$ has disconnected real locus $X(\mathbb{R})$ [Kra18, p.
117]; thus $X$ cannot be rational over $\mathbb{R}$. The cases $(d+2-e,e,1)$
with $2\leq e\leq\tfrac{d+2}{2}$ are connected.

###  5.3. Quadric $n$-folds over $\mathbb{R}$

Assume now that $X$ has even dimension $d=2n$. We may read off from the
invariant $(r_{1},\ldots,r_{2s+1})$ which classes of quadric $n$-folds
$Q\subset X_{\mathbb{C}}$ are realized by quadrics defined over $\mathbb{R}$.

Fix a smooth real quadric hypersurface

${\bf Q}=\\{F=0\\}\subset\mathbb{P}^{2n+1}.$

The following conditions are equivalent:

  * •

the geometric components of the variety of maximal isotropic subspaces
$\operatorname{OGr}(\bf{Q})$ are defined over $\mathbb{R}$;

  * •

the discriminant of $F$ is positive;

  * •

the $I^{+}(F)-I^{-}(F)$ is divisible by four.

This means that complex conjugation fixes the class of a maximal isotropic
subspace. We also have equivalence among:

  * •

there is a maximal isotropic subspace $\mathbb{P}^{n}\subset\bf{Q}$ defined
over $\mathbb{R}$;

  * •

the signature of $F$ is $(n+1,n+1)$.

Thus quadric $n$-folds

$Q\subset X\subset\mathbb{P}^{2n+2}$

defined over $\mathbb{R}$ correspond to rulings of degenerate fibers
$\mathcal{Q}_{t},t\in D(\mathbb{R})$ where $\mathcal{Q}_{t}$ has signature
$(n+1,n+1)$. As in Example 4.1, the corresponding $(+1)$-blocks in the complex
conjugation involution $\iota\in W(D_{2n+3})$ will be designated $\bf{1}$, in
boldface.

###  5.4. Analysis of the remaining even-dimensional isotopy classes

We continue to assume that $X$ has even dimension $d=2n$, focusing on the
isotopy classes without lines.

The cases

$(2n+2-e,e,1)=(e,1,2n+2-e),\quad 2\leq e\leq n+1$

have degeneracy consisting of $2n+3$ real points. The signatures of
nonsingular members are

$\displaystyle(1,2n+2)\ldots(e+1,2n+2-e)(e,2n+3-e)(e+1,2n+2-e)\ldots$
$\displaystyle(2n+1,2)(2n+2,1)\ldots(2n+2-e,e+1)(2n+3-e,e)$
$\displaystyle(2n+2-e,e+1)\ldots(2,2n+1)$

For $(n+1,n+1,1)$ the resulting signed permutation matrix is the diagonal
matrix

(5.1)
$\operatorname{diag}(\underbrace{(-1)^{n},\ldots,-1,{\mathbf{1}}}_{n+1\text{
terms }},{\mathbf{1}},\underbrace{{\mathbf{1}},-1,\ldots,(-1)^{n}}_{n+1\text{
terms }}),$

with the bolded $\bf{1}$’s corresponding to singular fibers with signature
$(n+1,n+1)$. The number of $+1$’s

$a=\begin{cases}n+2&\text{if $n$ odd }\\\ n+3&\text{if $n$ even.}\end{cases}$

For $e\neq n+1$ we have

$\operatorname{diag}(\underbrace{(-1)^{n},\ldots,(-1)^{n+e-1}}_{e\text{
terms}},(-1)^{n+e-1},\underbrace{(-1)^{n+(2n+2-e)-1},\ldots,(-1)^{n}}_{2n+2-e\text{
terms}}).$

Note that $(-1)^{n+e-1}=(-1)^{n+(2n+2-e)-1}$ so the three middle terms are
equal. There is exactly one $\bf{1}$ corresponding to the singular fiber with
signature $(n+1,n+1)$. The number of $+1$’s is given

(5.2) $a=\begin{cases}n&\text{if $n,e$ odd }\\\ n+2&\text{if $n$ odd and $e$
even}\\\ n+3&\text{if $n$ even and $e$ odd}\\\ n+1&\text{if $n,e$
even.}\end{cases}$

For case $(2n+1)$ the signatures of nonsingular members are

$\displaystyle(2,2n+1)\ldots(2n,3)(2n+1,2)(2n+2,3)\ldots(2,2n+1)$

The signed permutation matrix has one factor

$\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right)$

and diagonal entries

$((-1)^{n-1},\ldots,-1,1,-1,\ldots,(-1)^{n-1}).$

The number of positive terms is

$a=\begin{cases}n&\text{if $n$ odd}\\\ n+1&\text{if $n$ even.}\end{cases}$

##  6\. Application of the constructions

###  6.1. Proof of Theorem 1.1

Rationality is evident for isotopy classes of varieties that contain a line
defined over $\mathbb{R}$. Propostion 5.1 enumerates the remaining cases

$(5),(1,3,3),(1,2,4).$

These are covered by the following propositions.

######  Proposition 6.1.

Let $X\subset\mathbb{P}^{2n+2}$ be a smooth complete intersection of two
quadrics over $\mathbb{R}$ with invariant $(2n+1)$. Then $X$ is rational.

###### Proof.

The analysis in Section 5.4 indicates that complex conjugation exchanges two
classes of quadric $n$-folds associated to complex conjugate points in
$D(\mathbb{C})\setminus D(\mathbb{R})$. Denote these by $[Q]$ and $[\bar{Q}]$
– recall from (2.1) that

$[Q]\cdot[\bar{Q}]=1.$

Choosing a suitably general complex representation $Q\subset X_{\mathbb{C}}$,
the intersection $Q\cap\bar{Q}$ is proper. Then $Q\cap\bar{Q}$ consists of a
single rational point of $X$ with multiplicity one. In particular, the
hypotheses of Construction II are satisfied. ∎

######  Proposition 6.2.

Let $X\subset\mathbb{P}^{2n+2}$ be a smooth complete intersection of two
quadrics over $\mathbb{R}$ with invariant

$(2n+2-e,e,1)=(e,1,2n+2-e),\quad 2\leq e\leq n+1.$

Assume that either $e$ is even or $e=n+1$. Then $X$ is rational.

###### Proof.

Assume first that $e=n+1$. It follows from (5.1) in Section 5.4 that $X$
admits three classes

$\begin{array}[]{c|cccc}&h^{n}&Q_{1}&Q_{2}&Q_{3}\\\ \hline\cr h^{n}&4&2&2&2\\\
Q_{1}&2&1+(-1)^{n}&1&1\\\ Q_{2}&2&1&1+(-1)^{n}&1\\\
Q_{3}&2&1&1&1+(-1)^{n}\end{array}$

with each $Q_{i}$ realized by a quadric $n$-fold defined over $\mathbb{R}$.
Construction III gives rationality in this case.

Now assume that $e$ is even. The formula (5.2) shows that the numbers of
$+1$’s and $-1$’s appearing in the $\iota\in W(D_{2n+3})$ associated with
complex conjugation are as close as possible. If $n$ is even then we have
$n+1$ of the former and $n+2$ of the latter; when $n$ is odd we have $n+2$ of
the former and $n+1$ of the letter. Given an $n$-plane $P\subset
X_{\mathbb{C}}$, the formulas in Section 2.2 yield

$w_{P}\cdot w_{\bar{P}}=(-1)^{n+1}/4$

whence $P$ and $\bar{P}$ are disjoint in $X_{\mathbb{C}}$. Thus we may apply
Construction I to conclude rationality. ∎

######  Remark 6.3.

Remark 3.4 implies that $X$ does not admit curves (or surfaces!) of odd degree
defined over $\mathbb{R}$. These would force the existence of lines defined
over $\mathbb{R}$, which do not exist in this isotopy class.

###  6.2. Remaining six-dimensional case

To settle the rationality of six-dimensional complete intersections of two
quadrics $X\subset\mathbb{P}^{6}$, there is one remaining case in the Krasnov
classification: $(1,3,5)$.

The sequence of signatures of nonsingular elements of $\\{\mathcal{Q}_{t}\\}$
is:

$\displaystyle(1,8)(2,7)(3,6)(4,5)(5,4)(6,3)(5,4)(6,3)(7,2)$
$\displaystyle(8,1)(7,2)(6,3)(5,4)(4,5)(3,6)(4,5)(3,6)(2,7).$

The signed permutation is the diagonal matrix

$\operatorname{diag}(-1,1,-1,{\bf{1}},-1,-1,-1,1,-1)$

and the invariant cycles are:

$\begin{array}[]{c|cccc}&h^{3}&Q_{1}&Q_{2}&Q_{3}\\\ \hline\cr h^{3}&4&2&2&2\\\
Q_{1}&2&0&1&1\\\ Q_{2}&2&1&0&1\\\ Q_{3}&2&1&1&0\end{array}$

Here $Q_{1}$ corresponds to the singular fiber of signature $(4,4)$ and
$Q_{2}$ and $Q_{3}$ correspond to the singular fibers of signatures $(2,6)$
and $(6,2)$. If $P\subset X$ is a three-plane then $w_{P}\cdot
w_{\bar{P}}=-3/4$ and $P$ and $\bar{P}$ meet in a single point.

##  7\. Extensions and more general fields

We work over a field $k$ of characteristic zero. In this section, we give
further examples of rationality constructions for $2n$-dimensional
intersections of two quadrics over $k$, relying on special subvarieties of
dimension $n$.

###  7.1. Dimension four: intersection computations

Given $X\subset\mathbb{P}^{6}$, a smooth complete intersection of two
quadrics, we have

$\displaystyle c_{t}({\mathcal{T}}_{X})$
$\displaystyle\equiv(1+7ht+21h^{2}t^{2})/(1+2ht)^{2}\pmod{t^{3}}$
$\displaystyle\equiv(1+7ht+21h^{2}t^{2})(1-2ht+4h^{2}t^{2})^{2}\pmod{t^{3}}$
$\displaystyle\equiv 1+3ht+5h^{2}t^{2}\pmod{t^{3}}$

If $T\subset X$ is a smooth projective geometrically connected surface then

$\displaystyle c_{t}({\mathcal{N}}_{T/X})$
$\displaystyle=(1+3ht+5h^{2}t^{2})/(1-K_{T}t+\chi(T)t^{2})$
$\displaystyle=1+(3h+K_{T})t+(5h^{2}+3hK_{T}+K_{T}^{2}-\chi(T))t^{2},$

where $\chi(T)$ is the topological Euler characteristic. The expected
dimension of the deformation space of $T$ in $X$ is

$\displaystyle\chi({\mathcal{N}}_{T/X})$
$\displaystyle=2-\tfrac{1}{2}(3h+K_{T})K_{T}+\frac{1}{2}(3h+K_{T})^{2}$
$\displaystyle\hskip 142.26378pt-(5h^{2}+3hK_{T}+K_{T}^{2}-\chi(T))$
$\displaystyle=2\chi(\mathcal{O}_{T})+\tfrac{1}{2}(-h^{2}-3hK_{T})-K_{T}^{2}+\chi(T)).$

For example,

  * •

if $T=\mathbb{P}^{2}$ is embedded as a plane then $(T\cdot T)_{X}=2$ and $T$
is rigid;

  * •

if $T$ is a quadric then $(T\cdot T)_{X}=2$ and $T$ moves in a three-parameter
family;

  * •

if $T$ is a quartic scroll then $(T\cdot T)_{X}=6$ and moves in a five-
parameter family;

  * •

if $T$ is a sextic del Pezzo surface then $(T\cdot T)_{X}=12$ and $T$ moves in
an eight-parameter family.

###  7.2. Dimension four: surfaces with one apparent double point

Recall that Construction V gives the rationality of fourfolds admitting a
surface with one apparent double point. A classical result asserted by Severi
– see [CMR04, Th. 4.10] for a modern proof – characterizes smooth surfaces
$T\subset\mathbb{P}^{5}$ with one apparent double point, i.e., surfaces that
acquire one singularity on generic projection into $\mathbb{P}^{4}$:

  * •

$\deg(T)=4$: $T$ is a quartic scroll

$T\simeq\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2)^{2}),\quad\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(3));$

  * •

$\deg(T)=5$: $T$ is a quintic del Pezzo surface.

Thus Construction V says that a smooth complete intersection of two quadrics
$X\subset\mathbb{P}^{6}$ is rational if it contains a quartic scroll, a
quintic del Pezzo surface, or a sextic del Pezzo surface with a rational
point. Rationality always holds when there are positive-dimensional
subvarieties of odd degree (see Section 3.1), so we focus attention to the
first case.

We seek criteria for the existence of a quartic scroll $T\subset X$, defined
over $k$. Clearly, the class $[T]$ must be Galois-invariant; however, Galois-
invariant classes need not be represented by cycles over $k$.

The intersection computations above imply that

$[T]=[Q_{2}]+[Q_{3}],$

where $Q_{2}$ and $Q_{3}$ represent quadric surfaces in $X$, defined over the
algebraic closure. Assume that the class $[Q_{2}]+[Q_{3}]$ is Galois invariant
and represents algebraic cycles defined over the ground field. We look for
quartic scrolls $T\subset X$ with class $[T]=[Q_{2}]+[Q_{3}]$.

######  Remark 7.1.

Over $k=\mathbb{R}$, case $(1,2,4)$ has signed permutation

$\operatorname{diag}(1,-1,-1,-1,{\bf{1}},-1,1).$

The intersection form on the invariant part of
$H^{4}(X_{\mathbb{C}},\mathbb{Z})$ takes the form

$\begin{array}[]{c|cccc}&h^{2}&Q_{1}&Q_{2}&Q_{3}\\\ \hline\cr h^{2}&4&2&2&2\\\
Q_{1}&2&2&1&1\\\ Q_{2}&2&1&2&1\\\ Q_{3}&2&1&1&2\end{array}$

Here we assume $Q_{1}$ (and $h^{2}-Q_{1}$) are associated with the element of
$\\{\mathcal{Q}_{t}\\}$ with signature $(3,3)$ and $Q_{2}$ and $Q_{3}$ are
contributed by the elements with signatures $(1,5)$ and $(5,1)$. While $Q_{1}$
is definable over $\mathbb{R}$, $Q_{2}$ and $Q_{3}$ are not definable over
$\mathbb{R}$ (see Section 5.3).

The requisite real cycles exist in $[T]=[Q_{2}]+[Q_{3}]$. This follows from
the exact sequence in [CT15, §4], using the rationality of $X$ over
$\mathbb{R}$ to ensure the vanishing of the unramified cohomology. It would be
interesting to deduce this directly using cohomological machinery [Kah96].
However, we do not know whether such $X$ contain quartic scrolls, in general.

Let $M$ denote the moduli space of quartic scrolls in a fixed cohomology class
on $X$. There is a morphism

$\begin{array}[]{rcl}M&\rightarrow&(\mathbb{P}^{6})^{\vee}\\\
T&\mapsto&\operatorname{span}(T)\end{array}$

assigning to each scroll the hyperplane it spans.

Hyperplane sections $Y=H\cap X$ containing such scrolls are singular by the
Lefschetz hyperplane theorem. Computations in Macaulay2 indicate that a
generic such $Y$ has four ordinary singularities. If a complete intersection
of two quadrics $Y\subset\mathbb{P}^{5}$ contains a quartic scroll, it
contains two families of such scrolls, each parametrized by $\mathbb{P}^{3}$:
These arise from residual intersections in quadrics in

$I_{T}(2)/I_{Y}(2).$

Thus the residual family has class

$2h^{2}-[T]=(h^{2}-[Q_{2}])+(h^{2}-[Q_{3}]).$

The hyperplane sections of $X$ with four singularities should be parametrized
by a reducible surface with distinguished component
$\Sigma\subset(\mathbb{P}^{6})^{\vee}$.

We speculate that $\Sigma$ is a quartic del Pezzo surface, constructed as
follows: Consider the pencil of quadrics $\mathcal{Q}_{t}$ defining $X$ and
fix the pair of rank-six quadrics

$\mathcal{Q}_{t_{2}},\mathcal{Q}_{t_{3}}$

whose maximal isotropic subspaces sweep out $Q_{2}$ and $Q_{3}$. Let
$v_{i}\in\mathcal{Q}_{t_{i}}$ denote the vertices and $\ell$ the line they
span, which is defined over $k$ even when $t_{2}$ and $t_{3}$ are conjugate
over $k$. Projecting from $\ell$ gives a degree-four cover

$X\rightarrow\mathbb{P}^{4}.$

Geometrically, the covering group is the Klein four-group and the branch locus
consists of two quadric hypersurfaces $Y_{2},Y_{3}$ intersecting in a degree-
four del Pezzo surface $S_{23}$. Is $\Sigma\simeq S_{23}$ over $k$?

###  7.3. Dimension six: Threefolds with one apparent double point

Construction V indicates that the existence of a threefold $W\subset X$ with
one apparent double point yields rationality. The following classification
[CMR04] builds on constructions of Edge [Edg32]:

  * •

$\deg(W)=5$: a scroll in planes associated with two lines and twisted cubic,
or one line and two conics;

  * •

$\deg(W)=6$: an Edge variety constructed as a residual intersection

$Q\cap(\mathbb{P}^{1}\times\mathbb{P}^{3})=\Pi_{1}\cup\Pi_{2}\cup W,$

where $Q$ is a quadric hypersurface and the $\Pi_{i}\simeq\mathbb{P}^{3}$ are
fibers of the Segre variety under the first projection;

  * •

$\deg(W)=7$: an Edge variety constructed as a residual intersection

$Q\cap(\mathbb{P}^{1}\times\mathbb{P}^{3})=\Pi\cup W$

with the notation as above;

  * •

$\deg(W)=8$: a scroll in lines over $\mathbb{P}^{2}$ of the form
$\mathbb{P}(\mathcal{E})$ (one-dimensional quotients of $\mathcal{E}$) where
$\mathcal{E}$ is a rank-two vector bundle given as an extension

$0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{E}\rightarrow\mathcal{I}_{p_{1},\ldots,p_{8}}(4)\rightarrow
0$

for eight points $p_{1},\ldots,p_{8}\in\mathbb{P}^{2}$, no four collinear or
seven on a conic.

Given that rationality follows when there are positive-dimensional
subvarieties of odd degree (see Section 3.1) we focus on the varieties of even
degree.

###  7.4. Dimension six: Degree six Edge variety

Let $W\subset\mathbb{P}^{7}$ denote an Edge variety arising as follows:
Consider the Segre fourfold

$\mathbb{P}^{1}\times\mathbb{P}^{3}\subset\mathbb{P}^{7}$

and take the residual intersection to two copies of $\mathbb{P}^{3}$

$\\{0,\infty\\}\times\mathbb{P}^{3}\subset\mathbb{P}^{1}\times\mathbb{P}^{3}$

in a quadric hypersurface. The resulting threefold

$W\simeq\mathbb{P}^{1}\times\Sigma,$

where $\Sigma\subset\mathbb{P}^{3}$ is a quadric hypersurface. Note that the
ideal of $W\subset\mathbb{P}^{7}$ is generated by nine quadratic forms.
Complete intersections of two quadrics

$W\subset Y\subset\mathbb{P}^{7}$

depend on five parameters. A Magma computation shows that a generic such $Y$
has eight ordinary singularities.

Suppose we have an embedding $W\hookrightarrow X$, where
$X\subset\mathbb{P}^{8}$ is a smooth complete intersection of two quadrics.
For fixed $X$, the Hilbert scheme of such threefolds has dimension eight.
Realizing

$W\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$

we have

$\displaystyle c_{1}({\mathcal{N}}_{W/X})$ $\displaystyle=$ $\displaystyle
3(h_{1}+h_{2}+h_{3}),$ $\displaystyle c_{2}({\mathcal{N}}_{W/X})$
$\displaystyle=$ $\displaystyle 8(h_{1}h_{2}+h_{1}h_{3}+h_{2}h_{3}),$
$\displaystyle c_{3}({\mathcal{N}}_{W/X})$ $\displaystyle=$ $\displaystyle
4h_{1}h_{2}h_{3}.$

The Riemann-Roch formula gives $\chi(N_{W/X})=8$. Since $(W\cdot W)_{X}=4$,
the primitive class

$([W]-\tfrac{3}{2}h^{3})^{2}=4-18+9=-5.$

######  Remark 7.2.

Suppose that $X$ is defined over $\mathbb{R}$, and corresponds to the
$(1,2,6)$ case, with signed permutation matrix (see Section 5.4)

$\operatorname{diag}(-1,1,1,1,-1,{\bf{1}},-1,1,-1).$

The invariant cycles are:

$\begin{array}[]{c|cccccc}&h^{3}&Q_{1}&Q_{2}&Q_{3}&Q_{4}&Q_{5}\\\ \hline\cr
h^{3}&4&2&2&2&2&2\\\ Q_{1}&2&0&1&1&1&1\\\ Q_{2}&2&1&0&1&1&1\\\
Q_{3}&2&1&1&0&1&1\\\ Q_{4}&2&1&1&1&0&1\\\ Q_{5}&2&1&1&1&1&0\end{array}$

Here $Q_{1}$ corresponds to the singular fiber of signature $(4,4)$ and the
classes $Q_{2},\ldots,Q_{5}$ correspond to the singular fibers of signatures
$(2,6)$ and $(6,2)$.

The class

$[Q_{2}]+[Q_{3}]+[Q_{4}]+[Q_{5}]-[Q_{1}]$

has degree six and self-intersection four. Is it represented by cycles defined
over $\mathbb{R}$? Does it admit an Edge variety of degree six over
$\mathbb{R}$? Over more general $k$ where the requisite cycles exist?

###  7.5. Dimension six: Degree eight variety

Let $\mathcal{E}$ be a stable rank-two vector bundle on $\mathbb{P}^{2}$ with
invariants $c_{1}(\mathcal{E})=4L$ and $c_{2}(\mathcal{E})=8L^{2}$. Note that
$\Gamma(\mathcal{E})$ has dimension eight, giving an inclusion

$V:=\mathbb{P}(\mathcal{E}^{\vee})\subset\mathbb{P}^{7}.$

We have a tautological exact sequence

$0\rightarrow\mathcal{O}_{V}(-\xi)\rightarrow\mathcal{E}^{\vee}_{V}\rightarrow
Q\rightarrow 0,$

whence

$0\rightarrow Q(\xi)\rightarrow T_{V}\rightarrow T_{\mathbb{P}^{2}}\rightarrow
0.$

Thus we have the following

$\displaystyle\xi^{2}-4L\xi+8L^{2}$ $\displaystyle=0$ $\displaystyle
c(Q(\xi))$ $\displaystyle=1+(2\xi-4L)+(\xi^{2}-4L\xi+8L^{2}),$ $\displaystyle
c({\mathcal{T}}_{V})$ $\displaystyle=(1+3L+3L^{2})(c(Q(\xi))$
$\displaystyle=1+(2\xi-L)+(\xi^{2}-4L\xi+8L^{2}+6\xi L-12L^{2}+3L^{2})$

and we find

$\displaystyle c_{1}({\mathcal{N}}_{V/X})$ $\displaystyle=$ $\displaystyle
3\xi+L,$ $\displaystyle c_{2}({\mathcal{N}}_{V/X})$ $\displaystyle=$
$\displaystyle 5\xi^{2}-\xi L+2L^{2},$ $\displaystyle
c_{3}({\mathcal{N}}_{V/X})$ $\displaystyle=$ $\displaystyle
3\xi^{3}-3\xi^{2}L+2L^{2}\xi-9L^{3}.$

Note that

$\deg(L^{3})=0,\quad\deg(L^{2}\xi)=1,\quad\deg(L\xi^{2})=4,\text{ and
}\deg(\xi^{3})=8,$

so we conclude that

$(V\cdot V)_{X}=14.$

The primitive class $[V]-2h^{3}$ has self-intersection $14-4\cdot 8+16=-2$
which means that

$[V]=h^{3}+[Q_{2}]+[Q_{3}],\quad(Q_{2}\cdot Q_{3})=1,$

where $Q_{2}$ and $Q_{3}$ are classes of quadric threefolds in $X$, defined
over the algebraic closure. (Up to the action of the Weyl group $W(D_{9})$
this is the only possibility.)

Returning to the only remaining case in dimension six where rationality over
$\mathbb{R}$ remains open (see Section 6.2):

######  Question 7.3.

Let $X\subset\mathbb{P}^{8}$ be a smooth complete intersection of two quadrics
over $\mathbb{R}$ in isotopy class $(1,3,5)$. Which classes of codimension-
three cycles $X$ are realized over $\mathbb{R}$? Are there varieties with one
apparent double point, defined over $\mathbb{R}$, representing these classes?

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