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Alexis Kouvidakis

Publications and source records attributed to Alexis Kouvidakis.

18 recordsLinked to original sources
arXiv2023

The Fermat cubic and monodromy of lines

In this paper we study properties of the locus of second type lines of a general cubic threefold and fourfold. By analysing the geometry of the Fano scheme of lines of the Fermat cubic fourfold and in particular giving an explicit description of the locus of second type lines, we deduce that the Voisin map is birational over the second type locus. For a general cubic threefold, by studying properties of the second type locus again, we compute that various natural geometric monodromy groups are the full symmetric group.

math.AG↗
arXiv2023

Effective divisors on projectivized Hodge bundles and modular forms

We construct vector-valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular we construct basic modular forms for genus $2$ and $3$. We also discuss modular forms on the moduli of hyperelliptic curves. In that case the relative canonical bundle is a pull back of a line bundle on a ${\mathbb P}^1$-bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In an appendix we use our method to calculate divisor classes in the dual projectivized $k$-Hodge bundle determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-Zograf.

math.AG↗
arXiv2021

Geometry of lines on a cubic fourfold

For a general cubic fourfold $X\subset\mathbb{P}^5$ with Fano scheme of lines $F$, we prove a number of properties of the universal family of lines $I\to F$ and various subloci. We first describe the moduli and ramification theory of the genus four fibration $p:I\to X$ and explore its relation to a birational model of $F$ in $I$. The main part of the paper is devoted to describing the locus $V\subset F$ of triple lines, i.e., the fixed locus of the Voisin map $\phi:F\dashrightarrow F$, in particular proving it is an irreducible projective singular surface of class $21\mathrm{c}_2(\mathcal{U}_F)$ and detailing its intersection with the locus $S$ of second type lines. A consequence of the analysis of the singularities of $V$ is a geometric proof of the fact that if $X$ is very general, then the number of singular (necessarily 1-nodal) rational curves in $F$ of primitive class is 3780.

math.AG↗
arXiv2020

On some invariants of cubic fourfolds

For a general cubic fourfold $X \subset \mathbb{P}^5$, we compute the Hodge numbers of the locus $S \subset F$ of lines of second type. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any smooth cubic hypersurface.

math.AG↗
arXiv2018

Measures of irrationality of the Fano surface of a cubic threefold

For $X$ a smooth cubic threefold we study the Plücker embedding of the Fano surface of lines $S$ of $X$. We prove that if $X$ is general then the minimal gonality of a covering family of curves of $S$ is four and that this happens for a unique family of curves. The analysis also shows that there is a unique pentagonal connecting family of curves, which leads to the fact that the connecting gonality of $S$ is five whereas the degree of irrationality, i.e.\ the minimal degree of a rational map from $S$ to $\mathbb{P}^2$, is six.

math.AG↗
arXiv2017

Effective cycles on the symmetric product of a curve, I: the diagonal cone. (with an appendix by Ben Moonen)

In this paper and in its sequel [BKLV], we investigate the cone ${\rm Pseff}_n(C_d)$ of pseudoeffective $n$-cycles in the symmetric product $C_d$ of a smooth curve $C$. In the present paper, we study the convex-geometric properties of the cone generated by the $n$-dimensional diagonal cycles, which we call the $n$-dimensional diagonal cone. We prove that the $n$-dimensional diagonal cone is a perfect face of ${\rm Pseff}_n(C_d)$ along which ${\rm Pseff}_n(C_d)$ is locally finitely generated.

math.AG↗
arXiv2017

Effective cycles on the symmetric product of a curve, II: the Abel-Jacobi faces

In this paper, which is a sequel of [BKLV], we study the convex-geometric properties of the cone of pseudoeffective $n$-cycles in the symmetric product $C_d$ of a smooth curve $C$. We introduce and study the Abel-Jacobi faces, related to the contractibility properties of the Abel-Jacobi morphism and to classical Brill-Noether varieties. We investigate when Abel-Jacobi faces are non-trivial, and we prove that for $d$ sufficiently large (with respect to the genus of $C$) they form a maximal chain of perfect faces of the tautological pseudoeffective cone (which coincides with the pseudoeffective cone if $C$ is a very general curve).

math.AG↗
arXiv2016

The cycle classes of divisorial Maroni loci

We determine the cycle classes of effective divisors in the compactified Hurwitz spaces of curves of genus g with a linear system of degree d that extend the Maroni divisors on the open Hurwitz space. Our approach uses Chern classes associated to a global-to-local evaluation map of a vector bundle over a generic $P^1$-bundle over the Hurwitz space.

math.AG↗
arXiv2015

Divisors on Hurwitz spaces: an appendix to 'The cycle classes of divisorial Maroni loci'

The Maroni stratification on the Hurwitz space of degree $d$ covers of genus $g$ has a stratum that is a divisor only if $d-1$ divides $g$. Here we construct a stratification on the Hurwitz space that is analogous to the Maroni stratification, but has a divisor for all pairs $(d,g)$ with $d \leq g$ with a few exceptions and we calculate the divisor class of an extension of these divisors to the compactified Hurwitz space.

math.AG↗
arXiv2011

The Hodge bundle on Hurwitz spaces

In 2009 Kokotov, Korotkin and Zograf gave a formula for the class of the Hodge bundle on the Hurwitz space of admissible covers of genus g and degree d of the projective line. They gave an analytic proof of it. In this note we give an algebraic proof and an extension of the result.

math.AG↗
arXiv2011

Rational correspondences between moduli spaces of curves defined by Hurwitz spaces

By associating to a curve C of genus g=2k and a pencil of degree d=k+1 the so-called trace curve (resp. the reduced trace curve) we define a rational map from the Hurwitz space of admissible covers of genus g=2k and degree d=k+1 to a moduli space of stable curves. We study the induced map between the divisor class groups of these moduli spaces of curves.

math.AG↗
arXiv2010

The class of a Hurwitz divisor on the moduli of curves of even genus

We calculate the cycle class of the Hurwitz divisor $D_2$ on the moduli space of stable curves of genus $g=2k$ given by the degree $k+1$ covers of the projective line with simple ramification points, two of which lie in the same fibre. We also study some aspects of the geometry of the natural map of the Hurwitz space of admissible covers of degree $k+1$ and genus $2k$ to the moduli space of stable curves of genus $2k$.

math.AG↗
arXiv2009

A note on Fano surfaces of nodal cubic threefolds

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.

math.AG↗
arXiv2006

Cycle relations on Jacobian varieties

By using the Grothendieck-Riemann-Roch theorem we derive cycle relations modulo algebraic equivalence in the Jacobian of a curve. The relations generalize the relations found by Colombo and van Geemen and are analogous to but simpler than the relations recently found by Herbaut. In an appendix due to Zagier it is shown that these sets of relations are equivalent.

math.AG↗
arXiv1993

The automorphism group of the moduli space of semi stable vector bundles

Let ${\cal S}{\cal U}(r, L_0)$ denote the moduli space of semi stable vector bundles of rank $r$ and fixed determinant $L_0$ of degree $d$ on a smooth curve $C$ of genus $g \geq 3$. In this paper we describe the group of automorphisms of $ {\cal S}{\cal U}(r, L_0) $. The analogue of this result is carried out for the space ${\cal U}(r,d) $ of semi stable vector bundles of rark $r$ and degree $d$. As an application of the technics we use, we give in the appendix at the end of the paper a proof of the Torelli theorem for the moduli spaces ${\cal S}{\cal U}(r, L_0) $ for any rank $r$ and degree $d$.

alg-geom↗
arXiv1993

Picard groups of Hilbert schemes of curves

We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in ${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the classes of the generators in terms of some ``natural'' divisor classes.

alg-geom↗