Moduli Space Of K3 Surfaces Research Articles

Rational Points in the Noether-Lefschetz Locus of Moduli Spaces of K3 Surfaces

Abstract We give a characterization of rational points lying on the Noether-Lefschetz locus of moduli spaces of K3 surfaces by studying their lifting properties under some natural coverings of the ambient space. We then prove that the Bombieri-Lang conjecture implies non-density statements for the rational points in the Noether-Lefschetz locus, as predicted by a conjecture of Shafarevich.

Compact moduli of K3 surfaces with a nonsymplectic automorphism

We construct a modular compactification via stable slc pairs for the moduli spaces of K3 surfaces with a nonsymplectic group of automorphisms under the assumption that some combination of the fixed loci of automorphisms defines an effective big divisor, and prove that it is semitoroidal.

Explicit Computations with Cubic Fourfolds, Gushel–Mukai Fourfolds, and their Associated K3 Surfaces

We present some applications of the Macaulay2 software package SpecialFanoFourfolds, a package for working with Hodge-special cubic fourfolds and Hodge-special Gushel–Mukai fourfolds. In particular, we show how to construct new examples of such fourfolds, some of which turn out to be rational. We also describe how to calculate K3 surfaces associated with cubic or Gushel-Mukai fourfolds, which relies on an explicit unirationality of some moduli spaces of K3 surfaces.

On the irrationality of moduli spaces of K3 surfaces

We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genusgg. We prove that the growth is bounded by a polynomial function of degree14+ε14+\varepsilonfor anyε>0\varepsilon >0and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree1010. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated Hodge theoretically with certain cubic fourfolds, Gushel–Mukai fourfolds, and hyperkähler fourfolds.

Explicit constructions of K3 surfaces and unirational Noether–Lefschetz divisors

We provide methods to construct explicit examples of K3 surfaces. This leads to unirational constructions of Noether–Lefschetz divisors inside the moduli space of K3 surfaces of genus g. We implement Mukai's unirationality construction of the moduli spaces of K3 surfaces of genus g∈{6,…,10,12}, and we also present a new constructive proof of the unirationality of the moduli space of K3 surfaces of genus 11. Furthermore, we show the existence of three unirational hypersurfaces in any moduli space of K3 surfaces of genus g.

The ring of modular forms of $\mathrm{O}(2,4;\mathbb{Z})$ with characters

We show that the ring of modular forms with characters for $\mathrm{O}(2,4;\mathbb{Z})$ is generated by forms of weights 4, 4, 6, 8, 10, 10, 12, and 30 with three relations of weights 8, 20, and 60. The proof is based on the study of a moduli space of K3 surfaces.

Non-symplectic automorphisms of K3 surfaces with one-dimensional moduli space

The moduli space of K3 surfaces X with a purely non-symplectic automorphism \sigma of order n\geq 2 is one dimensional exactly when \varphi(n)=8 or 10 . In this paper we classify and give explicit equations for the very general members (X,\sigma) of the irreducible components of maximal dimension of such moduli spaces. In particular, we show that there is a unique one-dimensional component for n=20,22, 24 , three irreducible components for n=15 and two components in the remaining cases.

KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and$U(2)\oplus D_4^{\oplus 2}$lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of$\mathbb {P}^{1}\times \mathbb {P}^{1}$branched along a specific$(4,\,4)$curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient$(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$with the symmetric linearization.

The unirationality of the moduli space of K3 surfaces of genus 22

Using the connection discovered by Hassett between the Noether-Lefschetz moduli space mathcal {C}_{42} of special cubic fourfolds of discriminant 42 and the moduli space mathcal {F}_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over mathcal {F}_{22} is unirational.

Bott vanishing for algebraic surfaces

Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties. We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space M_{0,5}^bar of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz. In another direction, we prove sharp results on which K3 surfaces satisfy Bott vanishing. For example, a K3 surface with Picard number 1 satisfies Bott vanishing if and only if the degree is 20 or at least 24. For K3 surfaces of any Picard number, we have complete results when the degree is big enough. We build on Beauville, Mori, and Mukai's work on moduli spaces of K3 surfaces, as well as recent advances by Arbarello-Bruno-Sernesi, Ciliberto-Dedieu-Sernesi, and Feyzbakhsh. The most novel aspect of the paper is our analysis of what happens when there is an elliptic curve of low degree. (In other terminology, this concerns K3 surfaces that are unigonal, hyperelliptic, trigonal, or tetragonal.) It turns out that the crucial issue is whether an elliptic fibration has a certain special type of singular fiber.

Relations in the tautological ring of the moduli space of $K3$ surfaces

We study the interplay of the moduli of curves and the moduli of K3 surfaces via the virtual class of the moduli spaces of stable maps. Using Getzler’s relation in genus 1, we construct a universal decomposition of the diagonal in Chow in the third fiber product of the universal K3 surface. The decomposition has terms supported on Noether–Lefschetz loci which are not visible in the Beauville–Voisin decomposition for a fixed K3 surface. As a result of our universal decomposition, we prove the conjecture of Marian–Oprea–Pandharipande: the full tautological ring of the moduli space of K3 surfaces is generated in Chow by the classes of the Noether–Lefschetz loci. Explicit boundary relations are constructed for all \kappa classes. More generally, we propose a connection between relations in the tautological ring of the moduli spaces of curves and relations in the tautological ring of the moduli space of K3 surfaces. The WDVV relation in genus 0 is used in our proof of the MOP conjecture.

Tautological classes on moduli spaces of hyper-Kähler manifolds

In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of $h$-polarized hyperk\"ahler manifolds. Firstly, we construct the tautological ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3$^{[n]}$-type hyperk\"ahler manifolds with $n\leq 2$. Secondly, we prove the cohomological generalized Franchetta conjecture on universal family of these hyperk\"ahler manifolds.

Local to global principle for the moduli space of K3 surfaces

Recently S. Patrikis, J.F. Voloch, and Y. Zarhin have proven, assuming several well-known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, under some technical restrictions on the Picard rank. This is possible since abelian varieties and K3s are quite well described by ‘Hodge-theoretical’ results. In particular the theorem we present can be interpreted as follows: a family of ell -adic representations that looks like the one induced by the transcendental part of the ell -adic cohomology of a K3 surface (defined over a number field) determines a Hodge structure which in turn determines a K3 surface (which may be defined over a number field).

A vanishing result for tautological classes on the moduli of K3 surfaces

Looijenga's vanishing theorem on the moduli space of curves ${\mathcal M}_g$ says that the tautological ring vanishes above degree $g-2$. We prove an analogous result for the tautological cohomology ring of the moduli space of K3 surfaces.

Enhancements in F-theory models on moduli spaces of K3 surfaces with ADE rank 17

We study the moduli of elliptic K3 surfaces with a section with the $ADE$ rank 17. While the Picard number of a generic K3 surface in such moduli space is 19, the Picard number is enhanced to 20 at special points in the moduli. K3 surfaces become attractive K3 surfaces at these points. Either of the following two situations occurs at such special points: i) the Mordell-Weil rank of an elliptic K3 surface is enhanced, or ii) the gauge symmetry is enhanced. The first case i) is related to the appearance of a $U(1)$ gauge symmetry. In this note, we construct the moduli of K3 surfaces with $ADE$ types $E_7 D_{10}$ and $A_{17}$. We determine some of the special points at which K3 surfaces become attractive in the moduli of K3 surfaces with $ADE$ types $E_7 D_{10}$ and $A_{17}$. We investigate the gauge symmetries in F-theory compactifications on attractive K3 surfaces which correspond to such special points in the moduli times a K3 surface. $U(1)$ gauge symmetry arises in some F-theory compactifications.

Moduli spaces and Modular forms

The roots of both moduli spaces and modular forms go back to the theory of elliptic curves in the 19th century. Both topics have seen an enormous growth in the second half of the 20th century, but the interaction between the two remained limited. Recently there have been new developments that led to new points of contact between the two topics. One is the theory of K3 surfaces that is rapidly gaining a lot of new interest. Here the link with modular forms on orthogonal groups has led to progress on the Kodaira dimension of the moduli spaces of K3 surfaces. Another new development has been the use of moduli spaces of curves to gather new information about Siegel modular forms. The workshop intended to bring representatives from both the theory of moduli and the theory of modular forms together to further the interaction between the two topics as the time seemed ripe to do this.

K3 surfaces with a non-symplectic automorphism and product-quotient surfaces with cyclic groups

We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves C_1 \times C_2 by the diagonal action of either the group \mathbb Z/p\mathbb Z or the group \mathbb Z/2p\mathbb Z where p is an odd prime. These K3 surfaces admit a non-symplectic automorphism of order p induced by an automorphism of one of the curves C_1 or C_2 . We prove that most of the K3 surfaces admitting a non-symplectic automorphism of order p (and in fact a maximal irreducible component of the moduli space of K3 surfaces with a non-symplectic automorphism of order p ) are obtained in this way. In addition, we show that one can obtain the same set of K3 surfaces under more restrictive assumptions namely one of the two curves, say C_2 , is isomorphic to a rigid hyperelliptic curve with an automorphism \delta_p of order p and the automorphism of the K3 surface is induced by \delta_p . Finally, we describe the variation of the Hodge structures of the surfaces constructed and we give an equation for some of them.

Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

We analyze K3 surfaces admitting an elliptic fibration ℰ and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration ℰ/G comparing its properties to the ones of ℰ. We show that if ℰ admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group. Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.

The unirationality of the moduli spaces of 2-elementary K 3 surfaces

We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product, we describe configuration spaces of 5 ⩽d⩽ 8 points in ℙ2 as arithmetic quotients of type IV.

Symplectic Automorphisms and the Picard Group of a K3 Surface

We consider the symplectic action of a finite group G on a K3 surface X. The Picard group of X has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then investigate the classification of symplectic actions by a fixed finite group, using moduli spaces of K3 surfaces with symplectic G-action.