Search results for "Algebraic Geometry"
showing 10 items of 356 documents
p −1-Linear Maps in Algebra and Geometry
2012
At least since Habousch’s proof of Kempf’s vanishing theorem, Frobenius splitting techniques have played a crucial role in geometric representation theory and algebraic geometry over a field of positive characteristic. In this article we survey some recent developments which grew out of the confluence of Frobenius splitting techniques and tight closure theory and which provide a framework for higher dimension geometry in positive characteristic. We focus on local properties, i.e. singularities, test ideals, and local cohomology on the one hand and global geometric applicatioms to vanishing theorems and lifting of sections on the other.
On the minimal number of singular fibers with non-compact Jacobians for families of curves over P1
2016
Abstract Let f : X → P 1 be a non-isotrivial family of semi-stable curves of genus g ≥ 1 defined over an algebraically closed field k. Denote by s nc the number of the singular fibers whose Jacobians are non-compact. We prove that s nc ≥ 5 if k = C and g ≥ 5 ; we also prove that s nc ≥ 4 if char ( k ) > 0 and the relative Jacobian of f is non-smooth.
On the Betti numbers of three fat points in P1 × P1
2019
In these notes we introduce a numerical function which allows us to describe explicitly (and nonrecursively) the Betti numbers, and hence, the Hilbert function of a set Z of three fat points whose support is an almost complete intersection (ACI) in P1 × P1 . A nonrecursively formula for the Betti numbers and the Hilbert function of these configurations is hard to give even for the corresponding set of five points on a special support in P2 and we did not find any kind of this result in the literature. Moreover, we also give a criterion that allows us to characterize the Hilbert functions of these special set of fat points.
Infinitesimal deformations of double covers of smooth algebraic varieties
2003
The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. The space of all infinitesimal deformations has a representation as a direct sum of two subspaces. One is isomorphic to the space of simultaneous deformations of the branch locus and the base of the double covering. The second summand is the subspace of deformations of the double covering which induce trivial deformations of the branch divisor. The main result of the paper is a description of the effect of imposing singularities in the branch locus. As a special case we study deformations of Calabi--Yau threefolds which are non--singular models of do…
OPERADS AND JET MODULES
2005
Let $A$ be an algebra over an operad in a cocomplete closed symmetric monoidal category. We study the category of $A$-modules. We define certain symmetric product functors of such modules generalising the tensor product of modules over commutative algebras, which we use to define the notion of a jet module. This in turn generalises the notion of a jet module over a module over a classical commutative algebra. We are able to define Atiyah classes (i.e. obstructions to the existence of connections) in this generalised context. We use certain model structures on the category of $A$-modules to study the properties of these Atiyah classes. The purpose of the paper is not to present any really de…
Polarization types of isogenous Prym-Tyurin varieties
2007
Let p:C-->Y be a covering of smooth, projective curves which is a composition of ��:C-->C' of degree 2 and g:C'-->Y of degree n. Let f:X-->Y be the covering of degree 2^n, where the curve X parametrizes the liftings in C^{(n)} of the fibers of g:C'-->Y. Let P(X,��) be the associated Prym-Tyurin variety, known to be isogenous to the Prym variety P(C,C'). Most of the results in the paper focus on calculating the polarization type of the restriction of the canonical polarization of JX on P(X,��). We obtain the polarization type when n=3. When Y=P^1 we conjecture that P(X,��) is isomorphic to the dual of the Prym variety P(C,C'). This was known when n=2, we prove it when n=3, and…
Projective models of K3 surfaces with an even set
2006
The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient conditions to have an even set. Moreover we investigate their relation with K3 surfaces with a Nikulin involution.
New fourfolds from F-theory
2015
In this paper, we apply Borcea-Voisin's construction and give new examples of fourfolds containing a del Pezzo surface of degree six, which admit an elliptic fibration on a smooth threefold. Some of these fourfolds are Calabi-Yau varieties, which are relevant for the $N=1$ compactification of Type IIB string theory known as $F$-Theory. As a by-product, we provide a new example of a Calabi--Yau threefold with Hodge numbers $h^{1,1}=h^{2,1}=10$.
Moduli spaces of rank two aCM bundles on the Segre product of three projective lines
2016
Let P^n be the projective space of dimension n on an algebraically closed field of characteristic 0 and F be the image of the Segre embedding of P^1xP^1xP^1 inside P^7. In the present paper we deal with the moduli spaces of locally free sheaves E on F of rank 2 with h^i(F,E(t))=0 for i=1,2 and each integer t.
Embeddings of a family of Danielewski hypersurfaces and certain \C^+-actions on \C^3
2006
International audience; We consider the family of complex polynomials in \C[x,y,z] of the form x^2y-z^2-xq(x,z). Two such polynomials P_1 and P_2 are equivalent if there is an automorphism \varphi of \C[x,y,z] such that \varphi(P_1)=P_2. We give a complete classification of the equivalence classes of these polynomials in the algebraic and analytic category.