Michela Artebani

University of Concepción · Departamento de Matemática

In this paper we study the geometry of the 14 families of K3 surfaces of Picard number four with finite automorphism group, whose Neron—Severi lattices have been classified by È. B. Vinberg. We provide projective models, we identify the degrees of a generating set of the Cox ring and in some cases we prove the unirationality of the associated modul...

The moduli space of K3 surfaces X with a purely non-symplectic automorphism \sigma of order n\geq 2 is one dimensional exactly when \varphi(n)=8 or 10 . In this paper we classify and give explicit equations for the very general members (X,\sigma) of the irreducible components of maximal dimension of such moduli spaces. In particular, we show that t...

Let X be a projective K3 surface over C. We prove that its Cox ring has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form 2(f+f′), where f,f′ are classes of smooth elliptic curves with f⋅f′=2. This result and techniques using Koszul's type...

In this paper we study the geometry of the $14$ families of K3 surfaces of Picard number four with finite automorphism group, whose N\'eron-Severi lattices have been classified by \`E.B. Vinberg. We provide projective models, we identify the degrees of a generating set of the Cox ring and in some cases we prove the unirationality of the associated...

The moduli space of K3 surfaces $X$ with a purely non-symplectic automorphism $\sigma$ of order $n\geq 2$ is one dimensional exactly when $\varphi(n)=8$ or $10$. In this paper we classify and give explicit equations for the very general members $(X,\sigma)$ of the irreducible components of maximal dimension of such moduli spaces. In particular we s...

In this paper, we provide a complete classification of non-symplectic automorphisms of order 9 of complex K3 surfaces.

Let $X$ be a projective K3 surface over $\mathbb C$. We prove that its Cox ring $R(X)$ has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or of the form $2(f+f')$, where $f,f'$ are classes of elliptic fibrations with $f\cdot f'=2$. This result and tec...

In this paper we provide a complete classification of non-symplectic automorphisms of order nine of complex K3 surfaces.

We provide a combinatorial characterization of monomial linear systems on toric varieties whose general member is quasismooth. This is given both in terms of the Newton polytope and in terms of the matrix of exponents of a monomial basis.

We provide a combinatorial characterization of monomial linear systems on toric varieties whose general member is quasismooth. This is given both in terms of the Newton polytope and in terms of the matrix of exponents of a monomial basis.

A smooth complex projective curve is called pseudoreal if it is isomorphic to its conjugate but is not definable over the reals. Such curves, together with real Riemann surfaces, form the real locus of the moduli space $\mathcal M_g$. This paper deals with the classification of pseudoreal curves according to the structure of their automorphism grou...

A smooth complex projective curve is called pseudoreal if it is isomorphic to its conjugate but is not definable over the reals. Such curves, together with real Riemann surfaces, form the real locus of the moduli space $\mathcal M_g$. This paper deals with the classification of pseudoreal curves according to the structure of their automorphism grou...

In this article it is shown that S-Expansion procedure affects the geometry of a Lie group, changing it an leading us to the geometry of another Lie group with higher dimensionality. Is outlined, via an example, a method for determining the semigroup, which would provide a Lie algebra from another. Finally, it is proved that the Lie algebra obtaine...

In this article it is shown that S-expansion procedure affects the geometry of a Lie group, changing it and leading us to the geometry of another Lie group with higher dimensionality. A method for determining the semigroup, which would provide a Lie algebra from another, is outlined via an example. Finally, it is proved that a Lie algebra obtained...

Explicit examples of both, hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature. In this paper, we construct a tower of explicit examples of such kind of curves. In that tower there are both hyperelliptic curves and non-hyperelliptic curves.

Let $C$ be a smooth curve which is complete intersection of a quadric and a degree $k>2$ surface in $\mathbb{P}^3$ and let $C^{(2)}$ be its second symmetric power. In this paper we study the finite generation of the extended canonical ring $R(\Delta,K) := \bigoplus_{(a,b)\in\mathbb{Z}^2}H^0(C^{(2)},a\Delta+bK)$, where $\Delta$ is the image of the d...

We prove that the Borcea-Voisin mirror pairs of Calabi-Yau threefolds admit projective birational models that satisfy the Berglund-H\"ubsch-Chiodo-Ruan transposition rule. This shows that the two mirror constructions provide the same mirror pairs, as soon as both can be defined.

We provide a sufficient condition for a general hypersurface in a $\mathbb Q$-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-H\"ubsch-Krawitz construction in case the ambient is a $\mathbb Q$-Fano toric variety with torsion free class group and the defining polynom...

We study the Büchi K3 surface proving that it belongs to the one dimensional family of Kummer surfaces associated to genus two curves with automorphism group \(D_4\). We compute its Picard lattice and show that the rational points of the surface are Zariski-dense. Moreover, we provide analogous results for the Kummer surface associated to any genus...

In this paper we determine a minimal set of generators for the Cox rings of extremal rational elliptic surfaces. Moreover, we develop a technique for computing the ideal of relations between them which allows, in all but three cases, to provide a presentation of the Cox ring.

We study the geometry of B\"uchi's K3 surface showing that the rational points of this surface are Zariski-dense.

Let $X$ be a smooth projective algebraic curve of genus $g\geq 2$ defined over a field $K$. We show that $X$ can be defined over its field of moduli if it has odd signature, i.e. if the signature of the covering $X\to X/\Aut(X)$ is of type $(0;c_1,...,c_k)$, where some $c_i$ appears an odd number of times. This result is applied to $q$-gonal curves...

Let X be a hypersurface of a Mori dream space Z. We provide necessary and sufficient conditions for the Cox ring R(X) of X to be isomorphic to R(Z)/(f), where R(Z) is the Cox ring of Z and f is a defining section for X. We apply our results to Calabi-Yau hypersurfaces of toric Fano fourfolds. Our second application is to general degree d hypersurfa...

We prove that the mirror symmetry of Berglund-H\"ubsch-Chiodo-Ruan, applied to K3 surfaces with a non-symplectic involution, coincides with the mirror symmetry described by Dolgachev and Voisin.

We prove that the coarse moduli space of curves of genus six is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces.

We study automorphisms of order four on K3 surfaces. The symplectic ones have been first studied by Nikulin, they are known to fix six points and their action on the K3 lattice is unique. In this paper we give a classification of the purely non-symplectic automorphisms by relating the structure of their fixed locus to their action on cohomology, in...

We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3-surfaces of Picard number two, and explicitly compute the Cox rings of generic K3-surfaces with a non-symplectic in...

In this paper we investigate the relation between the finite generation of the Cox ring R(X) of a smooth projective surface X and its anticanonical Iitaka dimension k(-K_X). Comment: 15 pages, 2 figures. Section 4 revised and expanded

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...

S. Kondō defined a birational period map between the moduli space of genus three curves and a moduli space of polarized K3 surfaces. In this paper we give a resolution of the period map, providing a surjective morphism from a suitable compactification of $\mathcal{M}_{3}$ to the Baily-Borel compactification of a six dimensional ball quotient.

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.

We prove that the coarse moduli space of curves of genus 6 is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized 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.

This is a survey of the classical geometry of the Hesse configuration of 12 lines in the projective plane related to inflection points of a plane cubic curve. We also study two K3 surfaces with Picard number 20 which arise naturally in connection with the configuration.

The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a K3 surface with a non symplectic action of Z_4. In this paper we study the geometry of the one dimensional family of K3 surfaces associated to the locus of plane quartics with five nodes.

S. Kond\=o defined a birational period map from the moduli space of genus three curves to a moduli space of degree four polarized K3 surfaces. In this paper we extend the period map to a surjective morphism on a suitable compactification of $\mathcal M_3$ and describe its geometry.

S. Kond¯ o defined a birational period map from the moduli space of genus three curves to a moduli space of degree four polarized K3 surfaces. In this paper we extend the period map to a surjective morphism on a suitable compactification of M 3 and describe its geometry.

S. Kond\=o used periods of $K3$ surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given...

In this paper we give a birational model for the theta divisor of the intermediate Jacobian of a generic cubic threefold X . We use the standard realization of X as a conic bundle and a 4−dimensional family of plane quartics which are totally tangent to the discriminant quintic curve of such a conic bundle structure. The additional data of an even...

Let X X be a compact Riemann surface of genus g g and d ≥ 12 g + 4 d\geq 12g+4 be an integer. We show that X X admits meromorphic functions with monodromy group equal to the alternating group A d . A_d.

Let $X$ be a compact Riemann surface of genus $g$ and $d\geq 12g+4$ be an integer. We show that $X$ admits meromorphic functions with monodromy group equal to the alternating group $A_d.$