53 research outputs found
On the birational geometry of spaces of complete forms II: skew-forms
Moduli spaces of complete skew-forms are compactifications of spaces of
skew-symmetric linear maps of maximal rank on a fixed vector space, where the
added boundary divisor is simple normal crossing. In this paper we compute
their effective, nef and movable cones, the generators of their Cox rings, and
for those spaces having Picard rank two we give an explicit presentation of the
Cox ring. Furthermore, we give a complete description of both the Mori chamber
and stable base locus decompositions of the effective cone of some spaces of
complete skew-forms having Picard rank at most four.Comment: 16 page
On Comon's and Strassen's conjectures
Comon's conjecture on the equality of the rank and the symmetric rank of a
symmetric tensor, and Strassen's conjecture on the additivity of the rank of
tensors are two of the most challenging and guiding problems in the area of
tensor decomposition. We survey the main known results on these conjectures,
and, under suitable bounds on the rank, we prove them, building on classical
techniques used in the case of symmetric tensors, for mixed tensors. Finally,
we improve the bound for Comon's conjecture given by flattenings by producing
new equations for secant varieties of Veronese and Segre varieties.Comment: 12 page
Spherical blow-ups of Grassmannians and Mori Dream Spaces
In this paper we classify weak Fano varieties that can be obtained by
blowing-up general points in prime Fano varieties. We also classify spherical
blow-ups of Grassmannians in general points, and we compute their effective
cone. These blow-ups are, in particular, Mori dream spaces. Furthermore, we
compute the stable base locus decomposition of the blow-up of a Grassmannian in
one point, and we show how it is determined by linear systems of hyperplanes
containing the osculating spaces of the Grassmannian at the blown-up point, and
by the rational normal curves in the Grassmannian passing through the blown-up
point.Comment: 23 pages. Exposition improved and corrected a statement on the stable
base locus decomposition in Theorem 1.3 thanks to the comments of the refere
Varieties of sums of powers and moduli spaces of (1,7)-polarized abelian surfaces
We study the geometry of some varieties of sums of powers related to the
Klein quartic. This allows us to describe the birational geometry of certain
moduli spaces of abelian surfaces. In particular we show that the moduli space
of -polarized abelian surfaces with a
symmetric theta structure and an odd theta characteristic is unirational by
showing that it admits a dominant morphism from a unirational conic bundle.Comment: 14 page
Moduli of abelian surfaces, symmetric theta structures and theta characteristics
We study the birational geometry of some moduli spaces of abelian varieties
with extra structure: in particular, with a symmetric theta structure and an
odd theta characteristic. For a -polarized abelian surface, we show
how the parities of the influence the relation between canonical level
structures and symmetric theta structures. For certain values of and
, a theta characteristic is needed in order to define Theta-null maps. We
use these Theta-null maps and preceding work of other authors on the
representations of the Heisenberg group to study the birational geometry and
the Kodaira dimension of these moduli spaces.Comment: Final version. To appear in Commentarii Mathematici Helvetici (CMH
- …