arXiv subjects

Alice Garbagnati

Publications and source records attributed to Alice Garbagnati.

44 records · Page 3Linked to original sources

arXiv2010

New families of Calabi-Yau 3-folds without maximal unipotent monodromy

The aim of this paper is to construct families of Calabi--Yau 3-folds without boundary points with maximal unipotent monodromy and to describe the variation of their Hodge structures. In particular five families are constructed. In all these cases the variation of the Hodge structures of the Calabi--Yau 3-folds is basically the variation of the Hodge structures of a family of curves. This allows us to write explicitly the Picard--Fuchs equation for the 1-dimensional families. These Calabi--Yau 3-folds are desingularizations of quotients of the product of a (fixed) elliptic curve and a K3 surface admitting an automorphisms of order 4 (with some particular properties). We show that these K3 surfaces admit an isotrivial elliptic fibration.

math.AG↗

arXiv2009

Elliptic K3 surfaces with abelian and dihedral groups of symplectic automorphisms

We analyze K3 surfaces admitting an elliptic fibration $E$ and a finite group $G$ of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration $E/G$ comparing its properties to the ones of $E$. We show that if $E$ admits an $n$-torsion section, its quotient by the group of automorphisms induced by this section admits again an $n$-torsion section. Considering automorphisms coming from the base of the fibration, we can describe the Mordell--Weil lattice of a fibration described by Kloosterman. We give the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Griees and Lam. Moreover we show that for certain groups $H$ of $G$, $H$ subgroups of $G$, a K3 surface which admits $H$ as group of symplectic automorphisms actually admits $G$ as group of symplectic automorphisms.

math.AG↗

arXiv2009

The Picard-Fuchs equation of a family of Calabi-Yau threefolds without maximal unipotent monodromy

Recently J.C. Rohde constructed families of Calabi-Yau threefolds parametrised by Shimura varieties. The points corresponding to threefolds with CM are dense in the Shimura variety and, moreover, the families do not have boundary points with maximal unipotent monodromy. Both aspects are of interest for Mirror Symmetry. In this paper we discuss one of Rohde's examples in detail and we explicitly give the Picard-Fuchs equation for this one dimensional family.

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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.

math.AG↗

arXiv2008

A geometrical approach to Gordan--Noether's and Franchetta's contributions to a question posed by Hesse

Hesse claimed that an irreducible projective hypersurface in $\PP^n$ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved that this is true for $n\leq 3$ and constructed counterexamples for every $n\geq 4$. Gordan and Noether and Franchetta gave classification of hypersurfaces in $\PP^4$ with vanishing hessian and which are not cones. Here we translate in geometric terms Gordan and Noether approach, providing direct geometrical proofs of these results.

math.AG↗

arXiv2008

Symplectic Automorphisms on Kummer Surfaces

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface $X$ by a finite abelian group $G$ of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of $H^2(X, \Z)$ which is fixed by the isometries induced by $G$. However for certain groups these discriminants are not the same of those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.

math.AG↗

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.

math.AG↗

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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