arXiv subjects

Laure Flapan

Publications and source records attributed to Laure Flapan.

At least 19 recordsLinked to original sources

arXiv2026

Peskine sixfolds and Debarre-Voisin fourfolds with associated cubic fourfolds

We develop the notion of Peskine sixfolds with associated K3 surfaces and cubic fourfolds and work out numerical conditions for when these associations occur. In discriminant 24, the first family for which there is an associated cubic fourfold, we identify the cubic explicitly. Moreover, we prove that in this case the Fano variety of lines of the cubic fourfold is isomorphic to the associated Debarre-Voisin hyperkähler fourfold.

math.AG↗

arXiv2026

The geometry of antisymplectic involutions, II

We continue our study of fixed loci of antisymplectic involutions on projective hyper-Kähler manifolds of $\mathrm{K3}^{[n]}$-type induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice. We prove that if the divisibility of the ample class is 2, then one connected component of the fixed locus is a Fano manifold of index 3, thus generalizing to higher dimensions the case of the LLSvS 8-fold associated to a cubic fourfold. We also show that, in the case of the LLSvS 8-fold associated to a cubic fourfold, the second component of the fixed locus is of general type, thus answering a question by Manfred Lehn.

math.AG↗

arXiv2025

Extremal divisors on moduli spaces of K3 surfaces

We establish criteria for when Noether--Lefschetz divisors generate an extremal ray in the cone of pseudoeffective divisors of an orthogonal modular variety. In particular, we exhibit many extremal rays of the cone of pseudoeffective divisors on any moduli space~$\mathcal{F}_{2d}$ of quasi-polarized K3 surfaces of degree $d$, as well as on any normal projective $\mathbb{Q}$-factorial compactification $\overline{\mathcal{F}}_{2d}$ of $\mathcal{F}_{2d}$ lying over the Baily--Borel compactification.

math.AG↗

arXiv2025

Cones of Noether-Lefschetz divisors and moduli spaces of hyperkähler manifolds

We give a general formula for generators of the NL-cone, the cone of effective linear combinations of irreducible components of Noether-Lefschetz divisors, on an orthogonal modular variety. We then fully describe the NL-cone and its extremal rays in the cases of moduli spaces of polarized K3 surfaces and hyperkähler manifolds of known deformation type for low degree polarizations. Moreover, we exhibit explicit divisors in the boundary of NL-cones for polarizations of arbitrarily large degrees. Additionally, we study the NL-positivity of the canonical class for these modular varieties. As a consequence, we obtain uniruledness results for moduli spaces of primitively polarized hyperkähler manifolds of ${\rm{OG6}}$ and ${\rm{Kum}}_n$-type. Finally, we show that any family of polarized hyperkähler fourfolds of ${\rm{Kum}}_2$-type with polarization of degree $2$ and divisibility $2$ over a projective base is isotrivial.

math.AG↗

arXiv2025

Equivariant Kuznetsov components for cubic fourfolds with a symplectic involution

We study the equivariant Kuznetsov component $\mathrm{Ku}_G(X)$ of a general cubic fourfold $X$ with a symplectic involution. We show that $\mathrm{Ku}_G(X)$ is equivalent to the derived category $D^b(S)$ of a $K3$ surface $S$, where $S$ is given as a component of the fixed locus of the induced symplectic action on the Fano variety of lines on $X$.

math.AG↗

arXiv2024

Kodaira dimension of moduli spaces of hyperkähler varieties

We study the Kodaira dimension of moduli spaces of polarized hyperkähler varieties deformation equivalent to the Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional variety. This question was studied by Gritsenko-Hulek-Sankaran in the cases of $K3^{[2]}$ and OG10 type when the divisibility of the polarization is one. We generalize their results to higher dimension and divisibility. As a main result, for almost all dimensions $2n$ we provide a lower bound on the degree such that for all higher degrees, every component of the moduli space of polarized hyperkähler varieties of $K3^{[n]}$ type is of general type.

math.AG↗

arXiv2023

On product identities and the Chow rings of holomorphic symplectic varieties

For a moduli space $M$ of stable sheaves over a $K3$ surface $X$, we propose a series of conjectural identities in the Chow rings $CH_\star (M \times X^\ell),\, \ell \geq 1,$ generalizing the classic Beauville-Voisin identity for a $K3$ surface. We emphasize consequences of the conjecture for the structure of the tautological subring $R_\star (M) \subset CH_\star (M).$ The conjecture places all tautological classes in the lowest piece of a natural filtration emerging on $CH_\star (M)$, which we also discuss. We prove the proposed identities when $M$ is the Hilbert scheme of points on a $K3$ surface.

math.AG↗

arXiv2023

The geometry of antisymplectic involutions, I

We study fixed loci of antisymplectic involutions on projective hyperkähler manifolds of $\mathrm{K3}^{[n]}$-type. When the involution is induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice, we show that the number of connected components of the fixed locus is equal to the divisibility of the class, which is either 1 or 2.

math.AG↗

arXiv2021

Monodromy of Kodaira Fibrations of Genus $3$

A Kodaira fibration is a non-isotrivial fibration $f\colon S\rightarrow B$ from a smooth algebraic surface $S$ to a smooth algebraic curve $B$ such that all fibers are smooth algebraic curves of genus $g$. Such fibrations arise as complete curves inside the moduli space $\mathcal{M}_g$ of genus $g$ algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case $g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of $\mathcal{M}_3$ parametrizing curves whose Jacobians have extra endomorphisms.

math.AG↗

arXiv2021

Complete families of indecomposable non-simple abelian varieties

Given a fixed product of non-isogenous abelian varieties at least one of which is general, we show how to construct complete families of indecomposable abelian varieties whose very general fiber is isogenous to the given product and whose connected monodromy group is a product of symplectic groups or is a unitary group. As a consequence, we show how to realize any product of symplectic groups of total rank $g$ as the connected monodromy group of a complete family of $g'$-dimensional abelian varieties for any $g'\ge g$. These methods also yield a construction of a new Kodaira fibration with fiber genus $4$.

math.AG↗

arXiv2019

Period integrals and Hodge modules

We define a map $\mathcal{P}_M$ attached to any polarized Hodge module $M$ such that the restriction of $\mathcal{P}_M$ to a locus on which $M$ is a variation of Hodge structures induces the usual period integral pairing for this variation of Hodge structures. In the case that $M$ is the minimal extension of a simple polarized variation of Hodge structures $V$, we show that the homotopy image of $\mathcal{P}_M$ is the minimal extension of the graph morphism of the usual period integral map for $V$.

math.AG↗

arXiv2019

Geometry of Schreieder's varieties and some elliptic and K3 moduli curves

We study the geometry of a class of $n$-dimensional smooth projective varieties constructed by Schreieder for their noteworthy Hodge-theoretic properties. In particular, we realize Schreieder's surfaces as elliptic modular surfaces and Schreieder's threefolds as one-dimensional families of Picard rank $19$ $K3$ surfaces.

math.AG↗

arXiv2018

Some explicit elliptic modular surfaces

We consider algebraic surfaces, recently constructed by Schreieder, that are smooth models of the quotient of the self-product of a complex hyperelliptic curve by a $(\mathbb{Z}/3^c\mathbb{Z})$-action. We show that these surfaces are elliptic modular surfaces in the sense of Shioda, meaning in particular that they are universal families of explicit moduli of elliptic curves.

math.AG↗

arXiv2018

Chow motives associated to certain algebraic Hecke characters

Shimura and Taniyama proved that if $A$ is a potentially CM abelian variety over a number field $F$ with CM by a field $K$ linearly disjoint from F, then there is an algebraic Hecke character $λ_A$ of $K$ such that $L(A/F,s)=L(λ_A,s)$. We consider a certain converse to their result. Namely, let $A$ be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form $y^e=γx^f+δ$. Fix positive integers $a$ and $n$ such that $n/2 < a \leq n$. Under mild conditions on $e, f, γ, δ$, we construct a Chow motive $M$, defined over $F=\mathbb{Q}(γ,δ)$, such that $L(M/F,s)$ and $L(λ_A^a\barλ_A^{n-a},s)$ have the same Euler factors outside finitely many primes.

math.NT↗

arXiv2017

Unavoidable Sets of Partial Words of Uniform Length

A set X of partial words over a finite alphabet A is called unavoidable if every two-sided infinite word over A has a factor compatible with an element of X. Unlike the case of a set of words without holes, the problem of deciding whether or not a given finite set of n partial words over a k-letter alphabet is avoidable is NP-hard, even when we restrict to a set of partial words of uniform length. So classifying such sets, with parameters k and n, as avoidable or unavoidable becomes an interesting problem. In this paper, we work towards this classification problem by investigating the maximum number of holes we can fill in unavoidable sets of partial words of uniform length over an alphabet of any fixed size, while maintaining the unavoidability property.

cs.FL↗

arXiv2017

Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences

The Stern diatomic sequence is closely linked to continued fractions via the Gauss map on the unit interval, which in turn can be understood via systematic subdivisions of the unit interval. Higher dimensional analogues of continued fractions, called multidimensional continued fractions, can be produced through various subdivisions of a triangle. We define triangle partition-Stern sequences (TRIP-Stern sequences for short), higher-dimensional generalizations of the Stern diatomic sequence, from the method of subdividing a triangle via various triangle partition algorithms. We then explore several combinatorial results about TRIP-Stern sequences, which may be used to give rise to certain well-known sequences. We finish by generalizing TRIP-Stern sequences and presenting analogous results for these generalizations.

math.CO↗

arXiv2017

Hodge Groups of Hodge Structures with Hodge Numbers $(n,0,\ldots,0,n)$

This paper studies the possible Hodge groups of simple polarizable $\mathbb{Q}$-Hodge structures with Hodge numbers $(n,0,\ldots,0,n)$. In particular, it generalizes earlier work of Ribet and Moonen-Zarhin to completely determine the possible Hodge groups of such Hodge structures when $n$ is equal to $1$, $4$, or a prime $p$. In addition, the paper determines possible Hodge groups, under certain conditions on the endomorphism algebra, when $n=2p$, for $p$ an odd prime. A consequence of these results is that both the Hodge and General Hodge Conjectures hold for all powers of a simple $2p$-dimensional abelian variety satisfying the aforementioned conditions.

math.AG↗

arXiv2014

Cubic Irrationals and Periodicity via a Family of Multi-dimensional Continued Fraction Algorithms

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a, a-a^2) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u,u') with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u,u') has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. Thus these results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.

math.NT↗