K3 Surfaces Of Genus Research Articles

ACM bundles on a general K3 surface of degree 2

We shall study semi-stable aCM bundles on a K3 surface of degree 2 and Picard number 1.

Interpolation of fat points on K3 and abelian surfaces

AbstractWe prove that any number of general fat points of any multiplicities impose the expected number of conditions on a linear system on a smooth projective surface, in several cases including primitive linear systems on very general K3 and abelian surfaces, “Du Val” linear systems on blowups of at nine very general points, and certain linear systems on some ruled surfaces over elliptic curves. This is done by answering a question of the author about the case of only one fat point on a certain ruled surface, which follows from a circle of results due to Treibich–Verdier, Segal–Wilson, and others.

Contractions of hyper-Kähler fourfolds and the Brauer group

We study the geometry of exceptional loci of birational contractions of hyper-Kähler fourfolds that are of K3[2]-type. These loci are conic bundles over K3 surfaces and we determine their classes in the Brauer group. For this we use the results on twisted sheaves on K3 surfaces, on contractions and on the corresponding Heegner divisors. For a general K3 surface of fixed degree there are three (T-equivalence) classes of order two Brauer group elements. The elements in exactly two of these classes are represented by conic bundles on such fourfolds. We also discuss various examples of such conic bundles.

Explicit coverings of families of elliptic surfaces by squares of curves

We show that, for each $$n>0$$ , there is a family of elliptic surfaces which are covered by the square of a curve of genus $$2n+1$$ , and whose Hodge structures have an action by $${{\mathbb {Q}}}(\sqrt{-n})$$ . By considering the case $$n=3$$ , we show that one particular family of K3 surfaces are covered by the squares of curves of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in $${\mathbb {P}}^4$$ with 15 ordinary double points up to a map of finite degree of K3 surfaces. This gives an explicit proof of the Kuga–Satake–Deligne correspondence for these K3 surfaces and any K3 surfaces related to them by maps of finite degree, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.

Special Cubic Four-Folds, K3 Surfaces, and the Franchetta Property

Abstract O’Grady conjectured that the Chow group of 0-cycles of the generic fiber of the universal family over the moduli space of polarized K3 surfaces of genus $g$ is cyclic. This so-called generalized Franchetta conjecture has been solved only for low genera where there is a Mukai model (precisely, when $g\leq 10$ and $g=12, 13, 16, 18, 20$), by the work of Pavic–Shen–Yin. In this paper, as a non-commutative analogue, we study the Franchetta property for families of special cubic four-folds (in the sense of Hassett) and relate it to O’Grady’s conjecture for K3 surfaces. Most notably, by using special cubic four-folds of discriminant 26, we prove O’Grady’s generalized Franchetta conjecture for $g=14$, providing the first evidence beyond Mukai models.

Perfect points on curves of genus 1 and consequences for supersingular K3 surfaces

We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus 1 curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of genus 1 fibrations on supersingular K3 surfaces without purely inseparable multisections.

Birational geometry of the Mukai system of rank two and genus two

Using the techniques of Bayer–Macrì, we determine the walls in the movable cone of the Mukai system of rank two for a general K3 surface S of genus two. We study the (essentially unique) birational map to S^{[5]} and decompose it into a sequence of flops. We give an interpretation of the exceptional loci in terms of Brill–Noether loci.

Brill\u2013Noether general K3 surfaces with the maximal number of elliptic pencils of minimal degree

We explicitly construct Brill–Noether general K3 surfaces of genus 4, 6 and 8 having the maximal number of elliptic pencils of degrees 3, 4 and 5, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill–Noether general K3 surfaces of genus 4 and 6 without stable Lazarsfeld–Mukai bundles of minimal c_2.

Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties

In this paper, we discuss the problem of whether the difference $$[X]-[Y]$$ of the classes of a Fourier–Mukai pair (X, Y) of smooth projective varieties in the Grothendieck ring of varieties is annihilated by some power of the class $$\mathbb {L} = [ \mathbb {A}^1 ]$$ of the affine line. We give an affirmative answer for Fourier–Mukai pairs of very general K3 surfaces of degree 12. On the other hand, we prove that in each dimension greater than one, there exists an abelian variety such that the difference with its dual is not annihilated by any power of $$\mathbb {L}$$ , thereby giving a negative answer to the problem. We also discuss variations of the problem.

On the non-divisorial base locus of big and nef line bundles on K3[2]-type varieties

AbstractWe approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

Odd order obstructions to the Hasse principle on general K3 surfaces

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface $Y$ of degree 2 over $\mathbb{Q}$ together with a three torsion Brauer class $\alpha$ that is unramified at all primes except for 3, but ramifies at all 3-adic points of $Y$. Motivated by Hodge theory, the pair $(Y, \alpha)$ is constructed from a cubic fourfold $X$ of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for $\alpha$. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold $X$ (and hence the fibers) over $\mathbb{Q}_3$ and local solubility at all other primes.

Reconstruction of general elliptic K3 surfaces from their Gromov–Hausdorff limits

We show that a general elliptic K3 surface with a section is determined uniquely by its discriminant, which is a configuration of 24 points on the projective line. It follows that a general elliptic K3 surface with a section can be reconstructed from its Gromov-Hausdorff limit as the volume of the fiber goes to zero.

Nef Cones of Nested Hilbert Schemes of Points on Surfaces

AbstractLet X be the projective plane, a Hirzebruch surface, or a general K3 surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on X. We calculate the nef cone for two types of nested Hilbert schemes. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces.

On Polarized K3 Surfaces of Genus 33

AbstractWe prove that the moduli space of smooth primitively polarized K3 surfaces of genus 33 is unirational.

The Deligne-Mostow List and Special Families of Surfaces

We study whether there exist infinitely many surfaces with given discrete invariants for which the H^2 is of CM type. This is a surface analogue of a conjecture of Coleman about curves. We construct a large number of examples of families of surfaces with p_g = 1 that in the moduli space cut out a special subvariety; these provide a positive answer to our question. Most of these are families of K3 surfaces but we also obtain some families of surfaces of general type. As input for our construction we use the work of Deligne and Mostow. Finally we prove that a very general K3 surface cannot be dominated by a product of curves of small genus.

Moduli of nodal curves on K3 surfaces

We consider modular properties of nodal curves on general K3 surfaces. Let Kp be the moduli space of primitively polarized K3 surfaces (S,L) of genus p⩾3 and Vp,m,δ→Kp be the universal Severi variety of δ-nodal irreducible curves in |mL| on (S,L)∈Kp. We find conditions on p,m,δ for the existence of an irreducible component V of Vp,m,δ on which the moduli map ψ:V→Mg (with g=m2(p−1)+1−δ) has generically maximal rank differential. Our results, which for any p leave only finitely many cases unsolved and are optimal for m⩾5 (except for very low values of p), are summarized in Theorem 1.1 in the introduction.

SMOOTH, ISOLATED CURVES IN FAMILIES OF CALABI-YAU THREEFOLDS IN HOMOGENEOUS SPACES

We show the existence of smooth isolated curves of different degrees and genera in Calabi-Yau threefolds that are complete intersections in homogeneous spaces. Along the way, we classify all degrees and genera of smooth curves on BN general K3 surfaces of genus <TEX>${\mu}$</TEX>, where <TEX>$5{\leq}{\mu}{\leq}10$</TEX>. By results of Mukai, these are the K3 surfaces that can be realised as complete intersections in certain homogeneous spaces.

On k-gonal loci in Severi varieties on general K3 surfaces and rational curves on hyperkähler manifolds

In this paper we study the gonality of the normalizations of curves in the linear system |H| of a general primitively polarized complex K3 surface (S,H) of genus p. We prove two main results. First we give a necessary condition on p,g,r,d for the existence of a curve in |H| with geometric genus g whose normalization has a gdr. Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family of nodal curves in |H| of genus g carrying a gk1 and of dimension equal to the expected dimensionmin{2(k−1),g}. Relations with the Mori cone of the hyperkähler manifold Hilbk(S) are discussed.

A cohomological splitting criterion for locally free sheaves on arithmetically Cohen–Macaulay surfaces

AbstractIn this paper we obtain a cohomological splitting criterion for locally free sheaves on arithmetically Cohen–Macaulay surfaces with cyclic Picard group, which is similar to Horrocks' splitting criterion for locally free sheaves on projective spaces. We also recover a duality property which identifies a general K3 surface with a certain moduli space of stable sheaves on it, and obtain examples of stable, arithmetically Cohen–Macaulay, locally free sheaves of rank two on general surfaces of degree at least five in ${\mathbb P}^3$.

Density of rational curves on $$K3$$ surfaces

We proved that the union of rational curves is dense on a very general K3 surface and the union of elliptic curves is dense in the 1st jet space of a very general K3 surface, both in the strong topology.