We study the Zariski tangent cone TX π −→ X to an affine variety X and the closure TX of π−1(Reg(X)) in TX . We focus on the comparison between TX and TX , giving sufficient conditions on X in order that TX = TX . One aspect of the results is to understand when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Our other interest is to consider conditions ...

Batyrev has defined the stringy E-function for complex varieties with at most log terminal singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev defined its stringy Hodge numbers essentially as the coefficients of this E-function, generalizing the usual notion o...

This paper is devoted to the study of a certain class of principal bundles on del Pezzo surfaces, which were introduced and studied by Friedman and Morgan in [10]: The two authors showed that there exists a unique principal bundle (up to isomorphism) on a given (Gorenstein) del Pezzo surface satisfying certain properties. We call these bundles almost regular. In turn, we study them in families....

Abstract. Let X be a smooth scheme of finite type over a field K, let E be a locally free OX -bimodule of rank n, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor, HomGrA(−,−), on the category of graded right A-modules. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of HomGrA(OX ,−). When X is a smo...

We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skewdiagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties ...

Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison o varieties of codimension 2 in projective space. In this paper we study points in P and cuves in P in an attempt to see how far typical codimension 2 results will extend. While the results are satisfactory for small degree, we find in each case examples where we cannot dec...

Given a weighted flag variety wΣ(μ, u) corresponding to chosen fixed parameters μ and u, we present an algorithm to compute lists of all possible projectively Gorenstein n-folds, having canonical weight k and isolated orbifold points, appearing as weighted complete intersections in wΣ(μ, u) or some projective cone(s) over wΣ(μ, u). We apply our algorithm to compute lists of interesting classes ...

By associating a ‘motivic integral’ to every complex projective variety X with at worst canonical, Gorenstein singularities, Kontsevich [Kon95] proved that, when there exists a crepant resolution of singularities φ : Y → X, the Hodge numbers of Y do not depend upon the choice of the resolution. In this article we provide an elementary introduction to the theory of motivic integration, leading t...