# Kodaira dimension of moduli of special cubic fourfolds

@article{Tanimoto2019KodairaDO,
title={Kodaira dimension of moduli of special cubic fourfolds},
journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
year={2019}
}
• Published 4 September 2015
• Mathematics
• Journal für die reine und angewandte Mathematik (Crelles Journal)
<jats:p>A special cubic fourfold is a smooth hypersurface of degree 3 and dimension 4 that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether–Lefschetz divisors <jats:inline-formula id="j_crelle-2016-0053_ineq_9999"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0053_eq_0986.png" /> <jats:tex…
17 Citations

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## References

SHOWING 1-10 OF 35 REFERENCES

### Special Cubic Fourfolds

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

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

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

### New cubic fourfolds with odd degree unirational parametrizations

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

### The period map for cubic fourfolds

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

• Mathematics
Duke Mathematical Journal
• 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

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

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