arXiv subjects
Publications and source records attributed to Alessandra Sarti.
We introduce logarithmic Enriques varieties as a singular analogue of Enriques manifolds, generalizing the notion of log-Enriques surfaces introduced by Zhang. We focus mainly on the properties of the subfamily of log-Enriques varieties that admit a quasi-etale cover by a singular symplectic variety and we give many examples.
Enriques manifolds are non--simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. The goal of this note is to prove the Morrison--Kawamata cone conjecture for very general Enriques manifolds when the degree of the cover is prime. The proof uses the analogous result (established by Amerik--Verbitsky) for their universal cover. We also verify the conjecture for a very general Enriques manifold which is deformation equivalent to one of the known examples.
We give two explicit versions of the decomposition theorem of Beilinson, Bernstein and Deligne applied to the universal family of quartic surfaces of $\mathbb{P}^3$. The starting point of our investigation is the remark that the nodes of a quartic surface impose independent conditions to the linear system $\mid \mathcal{O}_{\mathbb P^3}(4)\mid$. Although this property is known in literature, we provide a different argument more suited to our purposes. By a result of \cite{DGF}, the independence of the nodes implies in turn that each component of Severi's variety is smooth of the expected dimension and that the dual variety is a divisor with normal crossings around Severi's variety. This allows us to study the complex $Rπ{_*}\mathbb{Q}_{\mathcal{X}}$, the derived direct image of the constant sheaf over the universal family of quartic surfaces $ \mathcal{X} \stackrelπ{\longrightarrow} \mathbb P^{34}$, both in the open set parametrizing smooth and nodal quartics and in a tubular neighborhood of the variety of Kummer surfaces. We obtain in both cases an explicit decomposition and a formality result for the complex $Rπ{_*}\mathbb{Q}_{\mathcal{X}}$.
A generalized Kummer surface $X$ of order $3$ is the minimal resolution of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism. We study a generalization of a problem of Shioda for classical Kummer surfaces, which is to understand how much $X$ is determined by $A$ and conversely. The surface $X$ posses a big and nef divisor $L_{X}$ such that $L_{X}^{2}=0$ or $2$ mod $6$. We show that for surfaces with $L_{X}^{2}=6k$ with $k\neq0,6\,mod\,9$, the surface $X$ determines the transcendental lattice $T(A)$ of $A$ and the Hodge structure on $T(A)$. Conversely if $A$ and $B$ are Fourier-Mukai partners (i.e. if the Hodge structures of their transcendental lattices are isomorphic) and $Y$ is the generalized Kummer surface which is the minimal resolution of the quotient of $B$ by an order $3$ symplectic automorphism, we obtain that $X$ and $Y$ are isomorphic. These results are also know to hold for surfaces with $L_{X}^{2}=2\,mod\,6$ from a previous work. When $k=0\text{ or }6\,mod\,9,$ we show that $X$ determines $T(A)$ and its Hodge structure, but the converse does not hold in general.
The (signed) projective cubes, as a special class of graphs closely related to the hypercubes, are on the crossroad of geometry, algebra, discrete mathematics and linear algebra. Defined as Cayley graphs on binary groups, they represent basic linear dependencies. Capturing the four-color theorem as a homomorphism target they show how mapping of discrete objects, namely graphs, may relate to special mappings of plane to projective spaces of higher dimensions. In this work, viewed as a signed graph, first we present a number of equivalent definitions each of which leads to a different development. In particular, the new notion of common product of signed graphs is introduced which captures both Cartesian and tensor products of graphs. We then have a look at some of their homomorphism properties. We first introduce an inverse technique for the basic no-homomorphism lemma, using which we show that every signed projective cube is of circular chromatic number 4. Then observing that the 4-color theorem is about mapping planar graphs into signed projective cube of dimension 2, we study some conjectures in extension of 4CT. Toward a better understanding of these conjectures we present the notion of extended double cover as a key operation in formulating the conjectures. With a deeper look into connection between some of these graphs and algebraic geometry, we discover that projective cube of dimension 4, widely known as the Clebsh graph, but also known as Greenwood-Gleason graph, is the intersection graph of the 16 straight lines of an algebraic surface known as Segre surface, which is a Del Pezzo surface of degree 4. We note that an algebraic surface known as the Clebsch surface is one of the most symmetric presentations of a cubic surface. Recall that each smooth cubic surface contains 27 lines. Hence, from hereafter, we believe, a proper name for this graph should be Segre graph.
We prove that the Fano variety of lines of a cuspidal cyclic cubic fourfold is a symplectic variety with transversal A2-singularities and we study the properties of the nonsymplectic order three automorphism induced by the covering automorphism on the irreducible holomorphic symplectic manifold obtained by blowing up the singular locus.
We study generalized Kummer surfaces Km$_{3}(A)$, by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism group. Such a surface carries $9$ disjoint configurations of two smooth rational curves $C,C'$ with $CC'=1$. This $9{\bf A}_{2}$-configuration plays a role similar to the Nikulin configuration of $16$ disjoint smooth rational curves on (classical) Kummer surfaces. We study the (generalized) question of T. Shioda: suppose that Km$_{3}(A)$ is isomorphic to Km$_{3}(B)$, does that imply that $A$ and $B$ are isomorphic? We answer by the negative in general, by two methods: by a link between that problem and Fourier-Mukai partners of $A$, and by construction of $9{\bf A}_{2}$-configurations on Km$_{3}(A)$ which cannot be exchanged under the automorphism group.
In this paper we present a classification of non-symplectic automorphisms of K3 surfaces whose order is a multiple of seven by describing the topological type of their fixed locus. In the case of purely non-symplectic automorphisms, we provide new results for order 14 and alternative proofs for orders 21, 28 and 42, so that we can unify in the same paper the results on these automorphisms. For each of these orders we also consider not purely non-symplectic automorphisms and obtain a complete characterization of their fixed loci. Several results of our paper were obtained independently in a recent paper by Brandhorst and Hofmann, but the methods used in the two papers are completely different.
We study some K3 surfaces obtained as minimal resolutions of quotients of subgroups of special reflection groups. Some of these were already studied in a previous paper by W. Barth and the second author. We give here an easy proof that these are K3 surfaces, give equations in weighted projective space and describe their geometry.
A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the $16\mathbf{A}_{1}$-configurations for usual Kummer surfaces. In this paper we construct $9$ other such $9\mathbf{A}_{2}$-configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $9\mathbf{A}_{2}$-configurations are obtained by taking the pullback of a certain configuration of $12$ conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of $X$ and of the generic fiber of its natural elliptic pencil.
We consider K3 surfaces of Picard rank 14 which admit a purely nonsymplectic automorphism of order 16. The automorphism acts on the second cohomology group with integer coefficients and we compute the invariant sublattice for the action. We show that all of these K3 surfaces admit an elliptic fibration and we compute the invariant lattices in a geometric way by using special curves of the elliptic fibration. The computation of these lattices plays an important role when one wants to study moduli spaces and mirror symmetry for lattice polarized K3 surfaces.
A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian surface from the Kummer structure is determined by the $16$ curves. A classical question of Shioda is about the existence of non isomorphic Kummer structures on the same Kummer K3 surface. The question was studied by several authors, and it was shown that the number of non-isomorphic Kummer structures is finite, but no explicit geometric construction of such structures was given. In a previous paper, we constructed explicitly non isomorphic Kummer structures on some Kummer surfaces. In this paper we generalise the construction to Kummer surfaces with a weaker restriction on the degree of the polarization and we describe some cases where the previous construction does not work.
We construct here many families of K3 surfaces that one can obtain as quotients of algebraic surfaces by some subgroups of the rank four complex reflection groups. We find in total 15 families with at worst $ADE$--singularities. In particular we classify all the K3 surfaces that can be obtained as quotients by the derived subgroup of the previous complex reflection groups. We prove our results by using the geometry of the weighted projective spaces where these surfaces are embedded and the theory of Springer and Lehrer-Springer on properties of complex reflection groups. This construction generalizes a previous construction by W. Barth and the second author.
It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M\_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M\_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\Km(E\_i\times E\_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,920$, both groups contains $M\_{20}$ as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part.
In this paper we classify non-symplectic automorphisms of order 8 on complex K3 surfaces in case that the fourth power of the automorphism has only rational curves in its fixed locus. We show that the fixed locus is the disjoint union of a rational curve and 10 isolated points or it consists in 4 isolated fixed points. We give examples corresponding to the case with a rational curve in the fixed locus and to the case with only isolated points in the fixed locus.
We consider compactifications of type IIA superstring theory on mirror-folds obtained as K3 fibrations over two-tori with non-geometric monodromies involving mirror symmetries. At special points in the moduli space these are asymmetric Gepner models. The compactifications are constructed from non-geometric automorphisms that arise from the diagonal action of an automorphism of the K3 surface and of an automorphism of the mirror surface. We identify the corresponding gaugings of N=4 supergravity in four dimensions, and show that the minima of the potential describe the same four-dimensional low-energy physics as the worldsheet formulation in terms of asymmetric Gepner models. In this way, we obtain a class of Minkowski vacua of type II string theory which preserve N=2 supersymmetry. The massless sector consists of N=2 supergravity coupled to 3 vector multiplets, giving the STU model. In some cases there are additional massless hypermultiplets.
A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration $\mathcal{C}$, then $X$ is a Kummer surface $X=Km(B)$ where $B$ is an Abelian surface determined by $\mathcal{C}$. Let $B$ be a generic Abelian surface having a polarization $M$ with $M^{2}=k(k+1)$ (for $k>0$ an integer) and let $X=Km(B)$ be the associated Kummer surface. To the natural Nikulin configuration $\mathcal{C}$ on $X=Km(B)$, we associate another Nikulin configuration $\mathcal{C}'$; we denote by $B'$ the Abelian surface associated to $\mathcal{C}'$, so that we have also $X=Km(B')$. For $k\geq2$ we prove that $B$ and $B'$ are not isomorphic. We then construct an infinite order automorphism of the Kummer surface $X$ that occurs naturally from our situation. Associated to the two Nikulin configurations $\mathcal{C},$ $\mathcal{C}'$, there exists a natural bi-double cover $S\to X$, which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for $k=2$ is a Schoen surface.
We investigate the interplay between the moduli spaces of ample <2>-polarized IHS manifolds of type K3^[2] and of IHS manifolds of type K3^[2] with a nonsymplectic involution with invariant lattice of rank one. In particular we geometrically describe some new involutions of the Hilbert square of a K3 surface, whose existence was proven in a previous work of Boissiere, Cattaneo, Nieper-Wisskirchen and Sarti.