K3 Surfaces Research Articles

Chow groups of surfaces of lines in cubic fourfolds

The surface of lines in a cubic fourfold intersecting a fixed line splits motivically into two parts, one of which resembles a K3 surface. We define the analogue of the Beauville-Voisin class and study the push-forward map to the Fano variety of all lines with respect to the natural splitting of the Bloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.

Higher Gaussian Maps on K3 Surfaces

Abstract We give sufficient conditions for the surjectivity of higher Gaussian maps on a polarized K3 surface. As an application, we show that the $k$-th Gaussian map for a general curve of genus $g$ (that depends quadratically with $k$) is surjective. Along the proof, we also exhibit an ampleness criterion for divisors in the Hilbert scheme of two points of a K3 surface.

Compact moduli of elliptic K3 surfaces

Compact moduli of elliptic K3 surfaces

Moduli of nodal sextic curves via periods of K3 surfaces

In this paper we study the moduli spaces of nodal sextic curves. We realize each irreducible component of the GIT space of sextic curves with given number of nodes as an open subspace of type IV arithmetic quotients. We then focus on the compactifications of the moduli spaces, one side is the geometric (GIT) compactifications, the other side is the Hodge theoretic compactifications such as Looijenga compactifications and Baily-Borel compactifications. The main result is the isomorphism between GIT and Looijenga compactifications. Some examples are closely related to del Pezzo surfaces. We also extend our results to moduli of nodal sextic curves with specified symmetry.

On K3 fibred Calabi–Yau threefolds in weighted scrolls

The aim of this paper is to classify mildly singular Calabi–Yau threefolds fibred in low-degree weighted K3 surfaces and embedded as anticanonical hypersurfaces in weighted scrolls, extending results of Mullet. We also study projective degenerations, revisiting an example due to Gross and Ruan. Finally we briefly discuss the general question of embedding a projective fibration into a weighted scroll.

The cotangent bundle of K3 surfaces of degree two

K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general polarised K3 surface $S$ of degree two. In particular, we describe the geometry of a surface $D_S \subset \mathbb{P}(\Omega_S)$ that plays a similar role to the surface of bitangents for a quartic in $\mathbb{P}^3$.

Lattice isometries and K3 surface automorphisms: Salem numbers of degree 20

This article extends Bayer-Fluckiger's theorem on characteristic polynomials of isometries on an even unimodular lattice to the case where the isometries have determinant −1. As an application, we show that the logarithm of every Salem number of degree 20 is realized as the topological entropy of an automorphism of a nonprojective K3 surface.

K3 surfaces with action of the group M20$M_{20}$

AbstractIt was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is 960 and if such a group has order 960, then it is isomorphic to the Mathieu group . In this paper, we are interested in projective K3 surfaces admitting a faithful symplectic action of the group . We show that there are infinitely many K3 surfaces with this action and we describe them and their projective models, giving some explicit examples.

A note on the Nielsen realization problem for 𝐾3 surfaces

We will show the following three theorems on the diffeomorphism and homeomorphism groups of a K 3 K3 surface. The first theorem is that the natural map π 0 ( D i f f ( K 3 ) ) → A u t ( H 2 ( K 3 ; Z ) ) \pi _{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb {Z})) has a section over its image. The second is that there exists a subgroup G G of π 0 ( D i f f ( K 3 ) ) \pi _{0}(Diff(K3)) of order two over which there is no splitting of the map D i f f ( K 3 ) → π 0 ( D i f f ( K 3 ) ) Diff(K3) \to \pi _{0}(Diff(K3)) , but there is a splitting of H o m e o ( K 3 ) → π 0 ( H o m e o ( K 3 ) ) Homeo(K3) \to \pi _{0}(Homeo(K3)) over the image of G G in π 0 ( H o m e o ( K 3 ) ) \pi _{0}(Homeo(K3)) , which is non-trivial. The third is that the map π 1 ( D i f f ( K 3 ) ) → π 1 ( H o m e o ( K 3 ) ) \pi _{1}(Diff(K3)) \to \pi _{1}(Homeo(K3)) is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for K 3 K3 surfaces.

Intersection theory on moduli of smooth complete intersections

We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus le 5 and polarized K3 surfaces of degree le 8.

The emergence proposal in quantum gravity and the species scale

In the Emergence Proposal in Quantum Gravity it is conjectured that all light-particle kinetic terms are absent in the fundamental ultraviolet theory and are generated by quantum corrections in the infrared. It has been argued that this may provide for some microscopic understanding of the Weak Gravity and Distance conjectures. In the present paper we take the first steps towards a systematic study of Emergence in the context of string theory. We emphasize the crucial role of the species scale in any effective field theory coupled to gravity, and discuss its computation in string theory and general systems with light towers of states. We then introduce the notion of Emergence and show how kinetic terms for moduli, gauge bosons and fermions may be generated. One-loop computations play an important role in Emergence, so we present detailed calculations in d spacetime dimensions for the wave-function renormalization of scalars, vectors and fermions. We extend and check the Emergence Proposal in a number string vacua, including 4d mathcal{N} = 2 theories arising from type IIA on a CY3, where the towers at strong coupling are comprised by D0 and (wrapped) D2-branes, and also elaborate on how instanton corrections would fit within the emergence picture. Higher dimensional examples are also discussed, including 6d and 7d models arising from F-/M-theory on an elliptic CY3 or a K3 surface. We also consider 10d string theories and study in some detail the emergence mechanism in type IIA. We show as well how the flux potential in 4d may be obtained from the emergence prescription, by analyzing the corresponding decompactification limits to M-theory. We find that the required kinetic terms for the dual 3-form fields can arise upon integrating out towers of massive gravitini (and bosonic superpartners). Our analysis renders support to the Emergence Proposal, and to the idea that infinite distance singularities may arise in Quantum Gravity as an intrinsic infrared phenomenon.

Geometric description of 〈2〉-polarised Hilbert squares of generic K3 surfaces

A generic K3 surface of degree 2t is a general complex projective K3 surface S2t whose Picard group is generated by the class of an ample divisor H∈Div(S2t) such that H2=2t with respect to the intersection form. We show that if X is the Hilbert square of a generic K3 surface of degree 2t with t≠2 which admits an ample divisor D∈Div(X) with qX(D)=2, where qX is the Beauville–Bogomolov–Fujiki form, then X is a double EPW sextic.

On Néron–Severi lattices of Jacobian elliptic K3 surfaces

On Néron–Severi lattices of Jacobian elliptic K3 surfaces

The cohomology of the general stable sheaf on a K3 surface

Let X be a K3 surface with Picard group Pic(X)≅ZH such that H2=2n. Let MH(v) be the moduli space of Gieseker semistable sheaves on X with Mukai vector v. We say that v satisfies weak Brill-Noether if the general sheaf in MH(v) has at most one nonzero cohomology group. We show that given any rank r≥2, there are only finitely many Mukai vectors of rank r on K3 surfaces of Picard rank one where weak Brill-Noether fails. We give an algorithm for finding the potential counterexamples and classify all such counterexamples up to rank 20 explicitly. Moreover, in each of these cases we calculate the cohomology of the general sheaf. Given r, we give sharp bounds on n, d, and a that guarantee that v satisfies weak Brill-Noether. As a corollary, we obtain another proof of the classification of Ulrich bundles on K3 surfaces of Picard rank one. In addition, we discuss the question of when the general sheaf in MH(v) is globally generated. Our techniques make crucial use of Bridgeland stability conditions.

Calabi–Yau threefolds with non‐Gorenstein involutions

AbstractThe concept of non‐Gorenstein involutions on Calabi–Yau threefolds is a higher dimensional generalization of non‐symplectic involutions on K3 surfaces. We present some elementary facts about Calabi–Yau threefolds with non‐Gorenstein involutions. We give a classification of the Calabi–Yau threefolds of Picard rank one with non‐Gorenstein involutions, whose fixed locus is not zero‐dimensional.

Appell series over finite fields and modular forms

In this article, we express the special values of F4⁎, a finite field analogue of the classical Appell series F4, in terms of the Fourier coefficients of weight three modular forms. These modular forms are related to modular K3 surfaces. To obtain these results, we first prove the finite field analogues of classical identities connecting classical F23-hypergeometric series and F4-Appell series.

A note on simple K3 singularities and families of weighted K3 surfaces

We discuss properties of the Seifert form for simple K3 singularities, and of the Picard lattices of families of weighted K3 surfaces. We study a collection $$\mathcal {M}_{(\rho ,\,\delta )}$$ of K3 surfaces polarized by their Picard lattices that are in the set $$\mathcal {L}_{(\rho ,\,\delta )}$$ of certain lattices. We also report a numerical formula that relates the Seifert form for the singularities and the Picard lattice of the family.

Explicit uniform bounds for Brauer groups of singular K3 surfaces

Let k be a number field. We give an explicit bound, depending only on [k:Q] and the discriminant of the Neron-Severi lattice, on the size of the Brauer group of a K3 surface X/k that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer-Manin set for such a variety is effectively computable. Conditional on GRH, we can also make the explicit bound depend only on [k:Q] and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on [k:Q], on the number of C-isomorphism classes of singular K3 surfaces defined over k, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.

Kernels of categorical resolutions of nodal singularities

AbstractIn this paper we study derived categories of nodal singularities. We show that for all nodal singularities there is a categorical resolution whose kernel is generated by a 2 or 3-spherical object, depending on the dimension. We apply this result to the case of nodal cubic fourfolds, where we describe the kernel generator of the categorical resolution as an object in the bounded derived category of the associated degree six K3 surface. This paper originated from one of the problem sessions at the Interactive Workshop and Hausdorff School “Hyperkähler Geometry”, Bonn, September 6–10, 2021.

Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces

We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer–Manin obstruction to the integral Hasse principle.