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The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular. The specialization induces an embedding of the N\'eron-Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point free involution that gives rise to the Enriques surface of type IV in Nikulin-Kondo-Martin's classification of Enriques surfaces with finite automorphism group. We investigate the specialization of this involution to characteristic 3., Comment: 32 pages. The proofs in Section 6 are simplified. The main results are not changed

We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj-Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103--108]., 8 pages,Mathematische Nachrichten

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., Comment: 27 pages

An ACM bundle on a polarized algebraic variety is defined as a vector bundle whose intermediate cohomology vanishes. We are interested in ACM bundles of rank one with respect to a very ample line bundle on a K3 surface. In this paper, we give a necessary and sufficient condition for a non-trivial line bundle $\mathcal{O}_X(D)$ on $X$ with $|D|=\emptyset$ and $D^2\geq L^2-6$ to be an ACM and initialized line bundle with respect to $L$, for a given K3 surface $X$ and a very ample line bundle $L$ on $X$., 17 pages. arXiv admin note: substantial text overlap with arXiv:1407.1703

For a trivial elliptic fibration $X=C \times S$ with $C$ an elliptic curve and $S$ a projective K3 surface of Picard rank $1$, we study how various notions of stability behave under the Fourier-Mukai autoequivalence $\Phi$ on $D^b(X)$, where $\Phi$ is induced by the classical Fourier-Mukai autoequivalence on $D^b(C)$. We show that, under some restrictions on Chern classes, Gieseker semistability on coherent sheaves is preserved under $\Phi$ when the polarisation is `fiber-like'. Moreover, for more general choices of Chern classes, Gieseker semistability under a `fiber-like' polarisation corresponds to a notion of $\mu_\ast$-semistability defined by a `slope-like' function $\mu_\ast$., Comment: 36 pages. Comments are welcome

We classifiy Enriques surfaces covered by the supersingular K3 surface with the Artin invariant 1 in characteristic 2. There are exactly three types of such Enriques surfaces., Comment: 32 pages, 4 figures; final version

Kollár (2008) introduced the surfaces ¶ ( x 1 a 1 x 2 + x 2 a 2 x 3 + x 3 a 3 x 4 + x 4 a 4 x 1 = 0 ) ⊂ P ( w 1 , w 2 , w 3 , w 4 ) ¶ where [math] , [math] , and [math] . The aim was to give many interesting examples of [math] -homology projective planes. They occur when [math] . For that case, we prove that Kollár surfaces are Hwang–Keum (2012) surfaces. For [math] , we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers [math] , where [math] are four general lines in [math] . In addition, by using various properties on classical Dedekind sums, we prove that: ¶ For any w ∗ > 1 , we have p g = 0 if and only if the Kollár surface is rational. This happens when a i + 1 ≡ 1 or a i a i + 1 ≡ − 1 ( mod w ∗ ) for some i . For any w ∗ > 1 , we have p g = 1 if and only if the Kollár surface is birational to a K3 surface. We classify this situation. For w ∗ ≫ 0 , we have that the smooth minimal model S of a generic Kollár surface is of general type with K S 2 ∕ e ( S ) → 1 .

We constructed several families of elliptic K3 surfaces with Mordell-Weil groups of ranks from 1 to 4. We studied F-theory compactifications on these elliptic K3 surfaces times a K3 surface. Gluing pairs of identical rational elliptic surfaces with nonzero Mordell-Weil ranks yields elliptic K3 surfaces, the Mordell-Weil groups of which have nonzero ranks. The sum of the ranks of the singularity type and the Mordell-Weil group of any rational elliptic surface with a global section is 8. By utilizing this property, families of rational elliptic surfaces with various nonzero Mordell-Weil ranks can be obtained by choosing appropriate singularity types. Gluing pairs of these rational elliptic surfaces yields families of elliptic K3 surfaces with various nonzero Mordell-Weil ranks. We also determined the global structures of the gauge groups that arise in F-theory compactifications on the resulting K3 surfaces times a K3 surface. $U(1)$ gauge fields arise in these compactifications., Comment: 25 pages

Using recent results of Bayer-Macr\`i, we compute in many cases the pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth projective K3 surface. We then use these results to construct explicit families of smooth curves on which the restriction of the cotangent bundle is not semistable (and hence not nef). In particular, this leads to a counterexample to a question of Campana-Peternell., Comment: Fixed some minor typos from the published version

We consider asymptotics of certain BPS state counts in M-theory compactified on a K3 surface. Our investigation is parallel to (and was inspired by) recent work in the mathematics literature by Filip, who studied the asymptotic count of special Lagrangian fibrations of a marked K3 surface, with fibers of volume at most $V_*$, in a generic twistor family of K3 surfaces. We provide an alternate proof of Filip's results by adapting tools that Douglas and collaborators have used to count flux vacua and attractor black holes. We similarly relate BPS state counts in 4d ${\cal N}=2$ supersymmetric gauge theories to certain counting problems in billiard dynamics and provide a simple proof of an old result in this field., Comment: 11 pages, 1 figure

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the DG algebra RHom(F,F) is formal for any sheaf F polystable with respect to an ample line bundle. Our main tool is the uniqueness of DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition., Comment: final version, to appear in Compos. Math

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).

Under the assumption of the Hodge, Tate and Fontaine-Mazur conjectures we give a criterion for a compatible system of l-adic representations to be isomorphic to the second cohomology of a K3 surface., Comment: 12 pages

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.

This thesis studies nodal sextics (algebraic curves of degree six), and in particular rational sextics, in the real projective plane. Two such sextics with k nodes are called rigidly isotopic if they can be joined by a path in the space of real nodal sextics with k nodes. The main result of the first part of the thesis is a rigid isotopy classification of real nodal sextics without real nodes, generalizing Nikulin’s classification of non-singular sextics. In the second part we study sextics with real nodes and we describe the rigid isotopy classes of such sextics in the case where the sextics are dividing, i.e., their real part separates the complexification (the set of complex points) into two halves. As a main application, we give a rigid isotopy classification for those nodal real rational sextics which can be perturbed to maximal or next-to-maximal sextics in the sense of Harnack’s inequality. Our approach is based on the study of periods of K3 surfaces, drawing on the Global Torelli Theorem by Piatetski-Shapiro and Shafarevich and Kulikov’s surjectivity theorem, as well as Nikulin’s results on symmetric integral bilinear forms., Cette thèse est consacrée à l’étude des courbes sextiques nodales, et en particulier des sextiques rationnelles, dans le plan projectif réel. Deux sextiques nodales réelles ayant k points doubles sont dites rigidement isotopes si elles peuvent être reliées par un chemin dans l’espace des sextiques nodales réelles ayant k points doubles. Le résultat principal de la première partie de la thèse donne une classification à isotopie rigide près des sextiques nodales irréductibles sans points doubles réels, généralisant la classification des sextiques non-singulières obtenue par Nikulin. La seconde partie porte sur les sextiques ayant des points doubles réels : une classification est obtenue pour les sextiques nodales séparantes, c’est-à-dire celles dont la partie réelle sépare leur complexification (l’ensemble des points complexes) en deux composantes connexes. Ce résultat est appliqué au cas des sextiques rationnelles réelles pouvant être perturbées en des sextiques maximales ou presque maximales (dans le sens de l’inégalité de Harnack). L’approche retenue repose sur l’étude des périodes des surfaces K3, se basant notamment sur le Théorème de Torelli Global de Piatetski-Shapiro et Shafarevich et le Théorème de Surjectivité de Kulikov, ainsi que sur les résultats de Nikulin portant sur les formes bilinéaires symétriques intégrales.

Let S be a K3 surface that admits a non-symplectic automorphism ρ of order 3. We divide S×P1 by ρ×ψ where ψ is an automorphism of order 3 of P1. There exists a ramified cover of a partial crepant resolution of the quotient that is a Calabi–Yau orbifold. We compute the Euler characteristic of our examples and obtain values ranging from 30 to 219.

We present a sufficient condition for the punctural Hilbert scheme of length two of a K3 surface with finite automorphism group to have automorphism group of infinite order in geometric terms (Theorem 2.1) and supply it with concrete examples (Theorem 1.2). We also discuss Mori dream space structures under an extermal crepant resolution (Theorems 1.2, 4.1, 5.2) from the viewpoint of automorphisms. These affirmatively answer a question of Malte Wandel.

In this article we consider moduli properties of singular curves on K3 surfaces. Let $\mathcal{B}_g$ denote the stack of primitively polarized K3 surfaces $(X,L)$ of genus $g$ and let $\mathcal{T}^n_{g,k} \to \mathcal{B}_g$ be the stack parametrizing tuples $[(f: C \to X, L)]$ with $f$ an unramified morphism which is birational onto its image, $C$ a smooth curve of genus $p(g,k)-n$ and $f_*C \in |kL|$. We show that the forgetful morphism $$\eta \; : \; \mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ is generically finite on one component, for all but finitely many values of $p(g,k)-n$. We further study the Brill--Noether theory of those curves parametrized by the image of $\eta$, and find a Wahl-type obstruction for a smooth curve with an unordered marking to have a nodal model on a K3 surface in such a way that the marking is the divisor over the nodes., Comment: Final version. To appear in J. Math. Pures. Appl

We describe pretty examples of derived equivalences and autoequivalences of Calabi-Yau threefolds arising from pencils of cubic fourfolds. The cubic fourfolds are chosen to be special, so they each have an associated K3 surface. Thus a pencil gives rise to two different Calabi-Yau threefolds: the associated pencil of K3 surfaces, and the baselocus of the original pencil—the intersection of two cubic fourfolds. They both have crepant resolutions which are derived equivalent.

Let Ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-vector space generated by the numbers given by all the periods ∫γΩ, γ ∈ H2(S, ℤ). We show that, if [Formula: see text], then S has complex multiplication, meaning that the Mumford–Tate group of the rational Hodge structure on H2(S, ℚ) is abelian. This result was announced in [P. Tretkoff, Transcendence and CM on Borcea–Voisin towers of Calabi–Yau manifolds, J. Number Theory 152 (2015) 118–155], without a detailed proof. The converse is already well known.

We show that the motive of the Hilbert scheme of length-$n$ subschemes on a K3 surface or on an abelian surface admits a decomposition similar to the decomposition of the motive of an abelian variety obtained by Shermenev, Beauville, and Deninger and Murre., 9 pages. To appear in Crelle's journal

We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if A and A ′ are Morita equivalent Azumaya algebras of degree r then 2r divides c 2 ( A ) − c 2 ( A ′ ) . In particular this implies that if A is an Azumaya algebra on a K3 surface and c 2 ( A ) is within 2r of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non-empty, is a proper algebraic space.

Recall that Ĉ ∈ |L = A d | is a nodal curvewith nodesN := {x1, . . . , xδ}on the polarized K3 surface (S,A ), such that A ∈ Pic(S) is not divisible, A 2 = 2(n − 1). (Note that the article [3] deals only with K3 surfaces with cyclic Picard group). Let σ : S → S be the blow-up of S at N, and denote by Ea , a = 1, . . . , δ, the exceptional divisors, and E := E1 + · · · + E . The normalization C of Ĉ fits into

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points., 9 pages. Minor changes; to appear in Crelle as an appendix to 1305.0178

We show that a characteristic \(0\) model \(X_R\rightarrow \mathrm{Spec \,}R\), with Picard number \(1\) over a geometric generic point, of a K3 surface in characteristic \(p\ge 3\), essentially kills all automorphisms (Theorem 5.1). We show that there is an explicitely constructed automorphism on a supersingular K3 surface in characteristic \(3\), which has positive entropy, the logarithm of a Salem number of degree \(22\) (Theorem 6.4). In particular it does not lift to characteristic \(0\). In addition, we show that in any large characteristic, there is an automorphism of a supersingular K3 which has positive entropy and does not lift to characteristic \(0\) (Theorem 7.5).

We prove a version of the Kawamata-Morrison ample cone conjecture for projective irreducible holomorphic symplectic manifolds deformation equivalent to either the Hilbert scheme of n points on a K3 surface, or a generalized Kummer variety.

We give a 1-dimensional family of classical and supersingular Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1. Moreover we show that there exist 30 nonsingular rational curves and ten non-effective (−2)-divisors on these Enriques surfaces whose reflection group is of finite index in the orthogonal group of the Neron-Severi lattice modulo torsion.

We construct and study curves with low H-constants on abelian and K3 surfaces. Using the Kummer $(16_{6})$-configurations on Jacobian surfaces and some $(16_{10})$-configurations of curves on $(1,3)$-polarized Abelian surfaces, we obtain examples of configurations of curves of genus $>1$ on a generic Jacobian K3 surface with H-constants $, Comment: Final version, 12 pages

Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index one or two in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur., 25 pages; final version, to appear in Compositio Mathematica

We extend the Kuga-Satake construction to the case of limit mixed Hodge structures of K3 type. We use this to study the geometry and Hodge theory of degenerations of Kuga-Satake abelian varieties, associated to polarized variations of K3 type Hodge structures over the punctured disc., 16 pages; to appear in J. Inst. Math. Jussieu

We study the good reduction modulo p of K3 surfaces with complex multiplication. If a K3 surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for K3 surfaces with Picard number 20. Our methods rely on the main theorem of complex multiplication for K3 surfaces by Rizov, an explicit description of the Breuil-Kisin modules associated with Lubin-Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral comparison theorem recently established by Bhatt, Morrow, and Scholze., 29 pages, to appear in International Mathematics Research Notices

Let F^N_g be the moduli space of polarized Nikulin surfaces (Y,H) of genus g and let P^N_g be the moduli of triples (Y,H,C), with C in |H| a smooth curve. We study the natural map χ_g:P^N_g -> R_g, where R_g is the moduli space of Prym curves of genus g. We prove that it is generically injective on every irreducible component, with a few exceptions in low genus. This gives a complete picture of the map χ_g and confirms some striking analogies between it and the Mukai map m_g: P_g ->M_g for moduli of triples (Y,H,C), where (Y,H) is any genus g polarized K3 surface. The proof is by degeneration to boundary points of a partial compactification of F^N_g. These represent the union of two surfaces with four even nodes and effective anticanonical class, which we call half Nikulin surfaces. The use of this degeneration is new with respect to previous techniques., 33 pages, comments are very welcome!

We construct a canonical basis of two-cycles, on a $K3$ surface, in which the intersection form takes the canonical form $2E_8(-1) \oplus 3H$. The basic elements are realized by formal sums of smooth submanifolds., 10 pages

We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,\xi,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $\xi$ and $\omega$ is a "generic" K\"ahler class on $S$, we show that the moduli space $M$ of $\mu_{\omega}-$stable sheaves on $S$ with associated Mukai vector $v$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface. If $M$ parametrizes only locally free sheaves, it is moreover hyperk\"ahler. Finally, we show that there is an isometry between $v^{\perp}$ and $H^{2}(M,\mathbb{Z})$ and that $M$ is projective if and only if $S$ is projective., Comment: 42 pages; major revisions; to appear in Kyoto J. Math

We analyze the structure of simply-connected Enriques surface in characteristic two whose K3-like covering is normal, building on the work of Ekedahl, Hyland and Shepherd-Barron. We develop general methods to construct such surfaces and the resulting twistor lines in the moduli stack of Enriques surfaces, including the case that the K3-like covering is a normal rational surface rather then a normal K3 surface. Among other things, we show that elliptic double points indeed do occur. In this case,there is only one singularity.The main idea is to apply flops to Frobenius pullbacks of rational elliptic surfaces, to get the desired K3-like covering. Our results hinge on Lang's classification of rational elliptic surfaces, the determination of their Mordell--Weil lattices by Shioda and Oguiso, and the behavior of unstable fibers under Frobenius pullback via Ogg's Formula. Along the way, we develop a general theory of Zariski singularities in arbitrary dimension, which is tightly interwoven with the theory of height-one group schemes actions and restricted Lie algebras. Furthermore, we determine under what conditions tangent sheaves are locally free, and introduce a theory of canonical coverings for arbitrary proper algebraic schemes., 61 pages. Mistake in Proposition 6.3 corrected, with related changes. Small rectifications in Section 15. Section 16 trimmed off. Otherwise minor modifications

In this paper, we prove the existence of an Enriques surface with a polarization of degree four with an Ulrich bundle of rank one. As a consequence, we prove that general polarized Enriques surfaces of degree four, with the same numerical polarization class, carry Ulrich line bundles., Comment: to appear in a volume of Ann. Univ. Ferrara dedicated to the memory of Alexandru Lascu

Let Y be a complex Enriques surface whose universal cover X is birational to a general quartic Hessian surface. Using the result on the automorphism group of X due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of Y. A fundamental domain of the action of the automorphism group on the nef cone of Y is described explicitly. The list of elliptic fibrations on Y and the list of combinations of rational double points that can appear on a surface birational to Y are presented. As an application, a set of generators of the automorphism group of the generic Enriques surface is calculated explicitly., It turns out that Theorem 1.6 of the previous version contains a mistake. We replace Theorem 1.6 with a corrected statement

Let $G$ be a finite abelian group which acts symplectically on a K3 surface. The N\'eron-Severi lattice of the projective K3 surfaces admitting $G$ symplectic action and with minimal Picard number is computed by Garbagnati and Sarti. We consider a $4$-dimensional family of projective K3 surfaces with $\mathbb{Z}_2^2$ symplectic action which do not fall in the above cases. If $X$ is one of these K3 surfaces, then it arises as the minimal resolution of a specific $\mathbb{Z}_2^3$-cover of $\mathbb{P}^2$ branched along six general lines. We show that the N\'eron-Severi lattice of $X$ with minimal Picard number is generated by $24$ smooth rational curves, and that $X$ specializes to the Kummer surface $\textrm{Km}(E_i\times E_i)$. We relate $X$ to the K3 surfaces given by the minimal resolution of the $\mathbb{Z}_2$-cover of $\mathbb{P}^2$ branched along six general lines, and the corresponding Hirzebruch-Kummer covering of exponent $2$ of $\mathbb{P}^2$., Comment: 24 pages, 6 figures. Final version with minor corrections and additions. To appear in the Rocky Mountain Journal of Mathematics

F-theory/heterotic duality is formulated in the stable degeneration limit of a K3 fibration on the F-theory side. In this note, we analyze the structure of the stable degeneration limit. We discuss whether stable degeneration exists for pairs of rational elliptic surfaces. We demonstrate that, when two rational elliptic surfaces have an identical complex structure, stable degeneration always exists. We provide an equation that systematically describes the stable degeneration of a K3 surface into a pair of isomorphic rational elliptic surfaces. When two rational elliptic surfaces have different complex structures, whether their sum glued along a smooth fiber admits deformation to a K3 surface can be determined by studying the structure of the K3 lattice. We investigate the lattice theoretic condition to determine whether a deformation to a K3 surface exists for pairs of extremal rational elliptic surfaces. In addition, we discuss the configurations of singular fibers under stable degeneration. The sum of two isomorphic rational elliptic surfaces glued together admits a deformation to a K3 surface, the singular fibers of which are twice that of the rational elliptic surface. For special situations, singular fibers of the resulting K3 surface collide and they are enhanced to a fiber of another type. Some K3 surfaces become attractive in these situations. We determine the complex structures and the Weierstrass forms of these attractive K3 surfaces. We also deduce the gauge groups in F-theory compactifications on these attractive K3 surfaces times a K3. $E_6$, $E_7$, $E_8$, $SU(5)$, and $SO(10)$ gauge groups arise in these compactifications., Comment: 24 pages, some clarifications

Let $S$ be a K3 surface and let $E$ be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold $S \times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for ellliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes., Comment: 68 pages

We study ACM bundles on cubic fourfolds containing a plane exploiting the geometry of the associated quadric fibration and Kuznetsov’s treatment of their bounded derived categories of coherent sheaves. More precisely, we recover the K3 surface naturally associated to the fourfold as a moduli space of Gieseker stable ACM bundles of rank four.

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant., Comment: 15 pages

Let X→ℙ 2 be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of characteristic diffierent from p. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut(X/ℙ 2) arise as pull-backs of certain Brauer classes on k(ℙ 2) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on k(ℙ 2) and relate the kernel of the pullback map to the Picard group of X. If p = 2, we give a second construction, which works over any base field of characteristic not 2, that uses Clifiord algebras arising from symmetric resolutions of line bundles on C to yield Azumaya representatives for the 2-torsion Brauer classes on X. We show that when −√−1 is in our base field, both constructions give the same result.

A generalized Kummer surface $X=Km(T,G)$ is the resolution of a quotient of a torus $T$ by a finite group of symplectic automorphisms $G$. We complete the classification of generalized Kummer surfaces by studying the two last groups which have not been yet studied. For these surfaces, we compute the associated Kummer lattice $K_{G}$, which is the minimal primitive sub-lattice containing the exceptional curves of the resolution $X\to T/G$. We then prove that a K3 surface is a generalised Kummer surface of type $Km(T,G)$ if and only if its N\'eron-Severi group contains $K_{G}$. For smooth-orbifold surfaces $\mathcal{X}$ of Kodaira dimension $\geq 0$, Kobayashi proved the orbifold Bogomolov Miyaoka Yau inequality $c_{1}^{2}(\mathcal{X})\leq3c_{2}(\mathcal{X}).$ For Kodaira dimension $2$, the case of equality is characterised as $\mathcal{X}$ being uniformized by the complex $2$-ball $\mathbb{B}_{2}$. For smooth-orbifold K3 and Enriques surfaces we characterize the case of equality as being uniformized by $\mathbb{C}^{2}$., Comment: 17 pages, Final version, Comments still welcome

In this article, we explicitly compute equations of an Enriques surface via the involution on a K3 surface. We also discuss its tropicalization and compute the tropical homology, thus recovering a special case of the result of [18], and establish a connection between the dimension of the tropical homology groups and the Hodge numbers of the corresponding algebraic Enriques surface.

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4-manifolds. Lower bounds on trisection genus are given in terms of Betti numbers and used to determine the trisection genus of all standard simply connected PL 4-manifolds. In addition, we construct trisections of small genus directly from the simplicial structure of triangulations using the Budney-Burton census of closed triangulated 4-manifolds. These experiments include the construction of minimal genus trisections of the non-orientable 4-manifolds $S^3 \tilde{\times} S^1$ and $\mathbb{R}P^4$., Comment: 16 pages, 2 figures, 5 tables