arXiv subjects

Alessandra Sarti

Publications and source records attributed to Alessandra Sarti.

52 records · Page 3Linked to original sources

arXiv2010

On symplectic and non--symplectic automorphisms of K3 surfaces

In this paper we investigate when the generic member of a family of K3 surfaces admitting a non--symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if the order of the automorphism is a prime number and we provide several examples and partial results otherwise. Moreover we prove that, under certain conditions, a K3 surface admitting a non--symplectic automorphism of prime odd order, $p$, also admits a non--symplectic automorphism of order $2p$. This generalizes a previous result by J. Dillies for $p=3$.

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arXiv2010

K3 surfaces with non-symplectic automorphisms of prime order

In this note we present the classification of non-symplectic automorphisms of prime order on K3 surfaces, i.e.we describe the topological structure of their fixed locus and determine the invariant lattice in cohomology. We provide new results for automorphisms of order 5 and 7 and alternative proofs for higher orders. Moreover, for any prime p, we identify the irreducible components of the moduli space of K3 surfaces with a non-symplectic automorphism of order p.

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arXiv2009

Elliptic fibrations and symplectic automorphisms on K3 surfaces

Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice $U^3\oplus E_8(-1)^2$ depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.

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arXiv2008

Symplectic automorphisms of prime order on K3 surfaces

The aim of this paper is to study algebraic K3 surfaces (defined over the complex number field) with a symplectic automorphism of prime order. In particular we consider the action of the automorphism on the second cohomology with integer coefficients. We determine the invariant sublattice and its perpendicular complement, and show that the latter coincides with the Coxeter-Todd lattice in the case of automorphism of order three. We also compute many explicit examples, with particular attention to elliptic fibrations.

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arXiv2008

Non-symplectic automorphisms of order 3 on K3 surfaces

In this paper we study K3 surfaces with a non-symplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli space and to show that it has three irreducible components.

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arXiv2008

On the Neron-Severi group of surfaces with many lines

For a binary quartic form $ϕ$ without multiple factors, we classify the quartic K3 surfaces $ϕ(x,y)=ϕ(z,t)$ whose Neron-Severi group is (rationally) generated by lines. For generic binary forms $ϕ$, $ψ$ of prime degree without multiple factors, we prove that the Neron-Severi group of the surface $ϕ(x,y)=ψ(z,t)$ is rationally generated by lines.

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arXiv2006

Projective models of K3 surfaces with an even set

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient conditions to have an even set. Moreover we investigate their relation with K3 surfaces with a Nikulin involution.

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arXiv2006

Counting lines on surfaces

This paper deals with surfaces with many lines. It is well-known that a cubic contains 27 of them and that the maximal number for a quartic is 64. In higher degree the question remains open. Here we study classical and new constructions of surfaces with high number of lines. We obtain in particular a symmetric octic with 352 lines.

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arXiv2006

Nikulin involutions on K3 surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the eleven dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison-Nikulin involutions, these are Nikulin involutions which interchange two copies of E_8(-1) in the Neron Severi group.

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arXiv2005

Trascendental Lattices of some K3-surfaces

In a previous paper, math.AG/0409419, we described six families of K3-surfaces with Picard-number 19, and we identified surfaces with Picard-number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover we show that the surfaces with Picard-number 19 are birational to a Kummer surface which is the quotient of a non-product type abelian surface by an involution.

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arXiv2005

Contraction of excess fibres between the McKay correspondences in dimensions two and three

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3-surfaces arising as minimal resolutions of quotients of nodal surfaces with high symmetries.

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arXiv2004

Group Actions, Cyclic Coverings and Families of K3-Surfaces

In this paper we describe six pencils of K3-surfaces which have large Picard-Number and contain precisely five singular fibers: four have A-D-E singularities and one is non-reduced. In particular we describe these surfaces as cyclic coverings of the K3-surfaces which Barth and the author described in a previous manuscript (Asian. J. of Math. Vol. 7, No. 4, 519-538, Dec. 2003). In many cases using this description and lattice-Theory we are able to compute the exact Picard-number and to describe explicitly the Picard-lattices.

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arXiv2004

A geometrical construction for the polynomial invariants of some reflection groups

In these notes we investigate the rings of real polynomials in four variables, which are invariant under the action of the reflectiongroups [3,4,3] and [3,3,5]. It is well known that they are rationally generated in degree 2,6,8,12 and 2,12,20,30. We give a different proof of this fact by giving explicit equations for the generating polynomials.

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arXiv2003

Polynomial Invariants of the Heisenberg Group with extra Symmetries

Let $G\subset SO(4)$ denote a finite subgroup containing the Heisenberg group. In these notes we classify all these groups, we find the dimension of the spaces of $G$-invariant polynomials and we give equations for the generators whenever the space has dimension two. Then we complete the study of the corresponding $G$-invariant pencils of surfaces in $\mathbb{P}_3$ which we started in math. AG/0106080. It turns out that we have five more pencils, two of them containing surfaces with nodes.

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arXiv2003

Polyhedral Groups and Pencils of K3-Surfaces with Maximal Picard Number

A K3-surface is a (smooth) surface which is simply connected and has trivial canonical bundle. In these notes we investigate three particular pencils of K3-surfaces with maximal Picard number. More precisely the general member in each pencil has Picard number 19 and each pencil contains four surfaces with Picard number 20. These surfaces are obtained as the minimal resolution of quotients X/G, where $G\subset SO(4,\R)$ is some finite subgroup and $X\subset ¶_3(\C)$ denotes a G-invariant surface. The singularities of X/G come from fix points of G on X or from singularities of X. In any case the singularities on X/G are A-D-E surface singularities. The rational curves which resolve them and some extra 2-divisible sets, resp. 3-divisible sets of rational curves generate the Neron-Severi group of the minimal resolution.

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arXiv2001

Pencils of symmetric surfaces in P_3

In this note we investigate three new pencils of symmetric surfaces in complex projective three-space. These have degree 6, 8 resp. 12 and are invariant under the action of subgroups of SO(4) containing the Heisenberg group. The pencils of degree 6 and 12 are invariant under the action of bigger groups, precisely under the action of the reflection groups F_4 resp. H_4. We compute equations for the generators of the pencils and describe the base locus of each pencil. We find also the singular surfaces and their number of singularities, which are, in fact, ordinary double points. In degree 12, we get a surface with 600 nodes. This confirms a conjecture of V. Goryounov and presents a new lower bound for the maximal number of nodes of such a surface.

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