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Samuel Boissière

Publications and source records attributed to Samuel Boissière.

13 recordsLinked to original sources
arXiv2023

The Fano variety of lines on singular cyclic cubic fourfolds

We study the symplectic resolution of the Fano variety of lines on some singular cyclic cubic fourfolds, i.e. cubic fourfolds arising as cyclic 3:1 cover of $\mathbb{P}^4$ branched along a cubic threefold. In particular we are interested in the geometry of these varieties in the case of cyclic cubic fourfolds branched along a cubic threefold having one isolated singularity of type $A_i$ for $i=2,3,4$. On these symplectic resolutions we find a non-symplectic automorphism of order three induced by the covering automorphism.

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arXiv2022

Some families of big and stable bundles on $K3$ surfaces and on their Hilbert schemes of points

Here we investigate meaningful families of vector bundles on a very general polarized $K3$ surface $(X,H)$ and on the corresponding Hyper--Kaehler variety given by the Hilbert scheme of points $X^{[k]}:= {\rm Hilb}^k(X)$, for any integer $k \geqslant 2$. In particular, we prove results concerning bigness and stability of such bundles. First, we give conditions on integers $n$ such that the twist of the tangent bundle of $X$ by the line bundle $nH$ is big and stable on~$X$; we then prove a similar result for a natural twist of the tangent bundle of $X^{[k]}$. Next, we prove global generation, bigness and stability results for tautological bundles on $X^{[k]}$ arising either from line bundles or from Mukai-Lazarsfeld bundles, as well as from Ulrich bundles on $X$, using a careful analysis on Segre classes and numerical computations for $k = 2, 3$.

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arXiv2021

Divisorial contractions to codimension three orbits

Let $G$ be a connected algebraic group. We study $G$-equivariant extremal contractions whose centre is a codimension three $G$-simply connected orbit. In the spirit of an important result by Kawakita in 2001, we prove that those contractions are weighted blow-ups.

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arXiv2020

Some algebraic and geometric properties of hyper-Kähler fourfold symmetries

We complete the classification of order $5$ nonsymplectic automorphisms on hyper-Kähler fourfolds deformation equivalent to the Hilbert square of a K3 surface. We then compute the topological Lefschetz number of natural automorphisms of generalized Kummer fourfolds and we describe the geometry of their fix loci.

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arXiv2019

Sur le produit de variétés localement factorielles ou Q-factorielles

We show that the factorial and Q-factorial loci of algebraic varieties defined over an algebraically closed field are open, that products of locally factorial varieties are still locally factorial, and that this property remains true for Q-factorial varieties if the ground field is not the algebraic closure of a finite field.

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arXiv2018

Cubic threefolds and hyperkähler manifolds uniformized by the 10-dimensional complex ball

We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6, both these spaces are uniformized by the same 10-dimensional arithmetic complex ball quotient. We then study the degeneration of the automorphism along the loci of nodal or chordal degenerations of the cubic threefold, showing the birationality of these loci with some moduli spaces of hyperkaehler fourfolds of K3^{[2]}-type with non-symplectic automorphism of order three belonging to different families. Finally, we construct a cyclic Pfaffian cubic fourfold to give an explicit construction of a non-natural automorphism of order three on the Hilbert square of a K3 surface.

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arXiv2015

Complex ball quotients from manifolds of $K3^{[n]}$-type

We describe periods of irreducible holomorphic manifolds of $K3^{[n]}$-type with a non-symplectic automorphism of prime order $p\geq 3$. These turn out to lie on complex ball quotients and we are able to give a precise characterization of when the period map is bijective, by introducing the notion of $K(T)$-generality.

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arXiv2015

On prime degree isogenies between K3 surfaces

We classify prime order isogenies between algebraic K3 surfaces whose rational transcendental Hodges structures are not isometric. The morphisms of Hodge structures induced by these isogenies are correspondences by algebraic classes on the product fourfolds; however, they do not satisfy the hypothesis of the well-known Mukai--Nikulin theorem. As an application we describe isogenies obtained from K3 surfaces with an action of a symplectic automorphism of prime order.

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arXiv2015

Borcea-Voisin Calabi-Yau threefolds and invertible potentials

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

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arXiv2014

Isometries of ideal lattices and hyperkähler manifolds

We prove that there exists a holomorphic symplectic manifold deformation equivalent to the Hilbert scheme of two points on a K3 surface that admits a non-symplectic automorphism of order 23, that is the maximal possible prime order in this deformation family. The proof uses the theory of ideal lattices in cyclomotic fields.

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arXiv2014

Classification of automorphisms on a deformation family of hyperkähler fourfolds by p-elementary lattices

We give a classification of all non-symplectic automorphisms of prime order p acting on irreducible holomorphic symplectic fourfolds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for p=2,3 and 7\leq p \leq 19. Our classification relates the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients with some invariants of the fixed locus and we provide explicit examples. As an application, we find new examples of non-natural non-symplectic automorphisms.

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