Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCox rings of cubic surfaces and Fano threefolds
142
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of Picard numbers one and two.
rate research
Read More
We give a definition of Cox rings and Cox sheaves for varieties over nonclosed fields that is compatible with torsors under quasitori, including universal torsors. We study their existence and classification, we make the relation to torsors precise, and we present arithmetic applications.
We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.
This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.
A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V otimes O_X by the sheaf of differentials Omega_X, given by the inclusion of a linear space V in Ext^1(O_X,Omega_X). For Lambda, a lattice of Cartier divisors, let R_Lambda denote the corresponding sheaf associated to V spanned by the first Chern classes of divisors in Lambda. We prove that any projective, smooth variety on which the bundle R_Lambda splits into a direct sum of line bundles is toric. We describe the bundle R_Lambda in terms of the sheaf of differentials on the characteristic space of the Cox ring, provided it is finitely generated. Moreover, we relate the finiteness of the module of sections of R_Lambda and of the Cox ring of Lambda.
We explore the connection between the rank of a polynomial and the singularities of its vanishing locus. We first describe the singularity of generic polynomials of fixed rank. We then focus on cubic surfaces. Cubic surfaces with isolated singularities are known to fall into 22 singularity types. We compute the rank of a cubic surface of each singularity type. This enables us to find the possible singular loci of a cubic surface of fixed rank. Finally, we study connections to the Hessian discriminant. We show that a cubic surface with singularities that are not ordinary double points lies on the Hessian discriminant, and that the Hessian discriminant is the closure of the rank six cubic surfaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
