arXiv subjects
Publications and source records attributed to Arnaud Beauville.
We discuss the notion of essential dimension of a finite group and explain its relation with birational algebraic geometry. We show how this leads to a (partial) classification of simple finite groups of essential dimension less than or equal to 3.
For a smooth complex projective variety, the rank of the Néron-Severi group is bounded by the Hodge number h^{1,1}. Varieties with rk NS = h^{1,1} have interesting properties, but are rather sparse, particularly in dimension 2. We discuss in this note a number of examples, in particular those constructed from curves with special Jacobians.
Let X be a reasonable topological space, G its fundamental group, and D = (G,G). We express the second quotient D/(D,G) of the lower central series of G in terms of the homology and cohomology of X . As an example, we recover the isomorphism D/(D,G) = Z/2 (due to Collino) when X is the Fano surface parametrizing lines in a cubic threefold.
We point out a link between two surfaces which have appeared recently in the literature: the surface of cuboids and the Schoen surface. Both surfaces give rise to a surface with q=4, whose canonical map is 2-to-1 onto an intersection of 4 quadrics in P^6 with 48 nodes.
The quartic hypersurfaces in P^4 invariant under the standard representation of S_6 form a linear pencil. We prove that a general member of this pencil is not rational.
The configuration of theta characteristics and vanishing thetanulls on a hyperelliptic curve is completely understood. We observe in this note that analogous results hold for the s-invariant theta characteristics on any curve C with an involution s . As a consequence we get examples of non hyperelliptic curves with a high number of vanishing thetanulls.
We associate to a unimodular lattice L, endowed with an automorphism of square -1, a principally polarized abelian variety A:= L_R/L. We show that the configuration of i-invariant theta divisors of A follows a pattern very similar to the classical theory of theta characteristics; as a consequence we find that A has a high number of vanishing thetanulls. When L = E_8 we recover the 10 vanishing thetanulls of the abelian fourfold discovered by R. Varley.
We give a simple proof of the non-rationality of the Fano threefold defined by the equations Σx_i = Σx_i^2 = Σx_i^3 = 0 in P^6 .
Let X be a holomorphic symplectic manifold, of dimension divisible by 4, and s an antisymplectic involution of X . The fixed locus F of s is a Lagrangian submanifold of X ; we show that its Â-genus is 1. As an application, we determine all possibilities for the Chern numbers of F when X is a deformation of the Hilbert square of a K3 surface.
Let A be a principally polarized abelian threefold over a perfect field k, not isomorphic to a product over the algebraic closure of k. There exists a canonical extension k' of k, of degree 1 or 2, such that A becomes isomorphic to a Jacobian over k'. The aim of this note is to give a geometric construction of this extension.
A list of open problems on holomorphic symplectic, contact and Poisson manifolds.
We describe the primitive cohomology lattice of a smooth even-dimensional complete intersection in projective space.
We classify, up to conjugacy, the finite subgroups of PGL(2,K) of order prime to char(K).
Let S be a complex Enriques surface; it is the quotient of a K3 surface X by a fixed-point-free involution. The Brauer group Br(S) has a unique nonzero element. We describe its pull-back in Br(X), and show that the surfaces S for which it is trivial form a countable union of hypersurfaces in the moduli space of Enriques surfaces.
We show that the algebraic group SL(2) acts on any polarized abelian variety A through correspondences. As a consequence we recover the action of SL(2) on the Chow group CH(A) (with rational coefficients), and this gives rise to Lefschetz type properties for algebraic cycles on A .
Let C be a general curve. Mukai asked whether for every stable rank 2 vector bundle E on C with det(E) = K, the multiplication map Sym^2 H^0(C,E) --> H^0(C, Sym^2 E) is injective. We observe that this holds when h^0(E) < 7 .
We classify, up to conjugacy, the subgroups of the Cremona group isomorphic to (Z/p)^r, where p is prime and r is maximal.
Let M be the moduli space of SO(r)-bundles on a curve, and L the determinant bundle on M. We define an isomorphism of H^0(M,L) onto the dual of the space of r-th order theta functions on the Jacobian of C. This isomorphism identifies the map M -->|L|* defined by the linear system |L| with the map M -->|r Theta| which associates to a quadratic bundle (E,q) the Theta divisor of the vector bundle E . The two components M+ and M- of M are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous question for Sp(2r)-bundles.