# Kodaira dimension of moduli of special cubic fourfolds

@article{Tanimoto2019KodairaDO, title={Kodaira dimension of moduli of special cubic fourfolds}, author={Sho Tanimoto and Anthony V{\'a}rilly-Alvarado}, journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)}, year={2019} }

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## 17 Citations

### Kodaira dimension of moduli of special $K3^{[2]}$-fourfolds of degree 2

- Mathematics
- 2019

We study the Noether-Lefschetz locus of the moduli space $\mathcal{M}$ of $K3^{[2]}$-fourfolds with a polarization of degree $2$. Following Hassett's work on cubic fourfolds, Debarre, Iliev, and…

### On the period map for polarized hyperk\"ahler fourfolds

- Mathematics
- 2017

This is an improved version of the eprint previously entitled "Unexpected isomorphisms between hyperkahler fourfolds."
We study smooth projective hyperkahler fourfolds that are deformations of…

### Unirationality of Certain Universal Families of Cubic Fourfolds

- MathematicsRationality of Varieties
- 2021

The aim of this short note is to define the \it universal cubic fourfold \rm over certain loci of their moduli space. Then, we propose two methods to prove that it is unirational over the Hassett…

### ABELIAN $n$ -DIVISION FIELDS OF ELLIPTIC CURVES AND BRAUER GROUPS OF PRODUCT KUMMER & ABELIAN SURFACES

- MathematicsForum of Mathematics, Sigma
- 2017

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$ . In 2008, Skorobogatov and Zarhin showed that the Brauer group…

### On the moduli space of pairs consisting of a cubic threefold and a hyperplane

- MathematicsAdvances in Mathematics
- 2018

### Hyperk\"ahler manifolds

- Mathematics
- 2018

The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperkahler manifolds. These manifolds are…

### On the Kodaira dimension of orthogonal modular varieties

- Mathematics
- 2017

We prove that up to scaling there are only finitely many integral lattices L of signature (2, n) with $$n\ge 21$$n≥21 or $$n=17$$n=17 such that the modular variety defined by the orthogonal group of…

### Trisecant flops, their associated K3 surfaces and the rationality of some cubic fourfolds

- MathematicsJournal of the European Mathematical Society
- 2022

We prove the rationality of some Fano fourfolds via Mori Theory and the Minimal Model Program. The method shows a connection between some admissible cubic fourfolds and some birational models of…

### The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

- MathematicsJournal of the London Mathematical Society
- 2021

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, U⊕⟨−2k⟩ ‐polarized K3 surfaces. Such moduli spaces are proved to be of general type for k⩾220 . The proof…

### Counting points on K3 surfaces and other arithmetic-geometric objects

- Mathematics
- 2018

This PhD thesis concerns
the topic of arithmetic geometry. We address three different questions and each
of the questions in some way is about counting how big some set is or can be.
We produce…

## References

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- MathematicsCompositio Mathematica
- 2000

AbstractA cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countably…

### The K3 category of a cubic fourfold

- MathematicsCompositio Mathematica
- 2017

Smooth cubic hypersurfaces $X\subset \mathbb{P}^{5}$ (over $\mathbb{C}$ ) are linked to K3 surfaces via their Hodge structures, due to the work of Hassett, and via a subcategory…

### Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

- Mathematics
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We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational…

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- Mathematics
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We prove that the moduli space of cubic fourfolds $\mathcal{C}$ contains a divisor $\mathcal{C}_{42}$ whose general member has a unirational parametrization of degree 13. This result follows from a…

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- Mathematics
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The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett)…

### Hodge theory and derived categories of cubic fourfolds

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- 2014

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with…

### The Kodaira dimension of the moduli of K3 surfaces

- Mathematics
- 2007

The global Torelli theorem for projective K3 surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treating moduli problems for K3 surfaces. The moduli…

### Brauer Groups on K3 Surfaces and Arithmetic Applications

- Mathematics
- 2017

For a prime p, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to…

### The moduli space of cubic fourfolds via the period map

- Mathematics
- 2007

We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the GIT…

### An Explicit Formula for Local Densities of Quadratic Forms

- Mathematics
- 1998

Let S and T be two positive definite integral matrices of rank m and n respectively. It is an ancient but still very challenging problem to determine how many times S can represent T , i.e., the…