Let $ e_1, ..., e_c be positive integers and let Y \subseteq \mathbb{P}^n$ the monomial complete intersection defined by vanishing of $x_1^{e_1}, x_c^{e_c}$. In this paper we study sharp upper bounds on number equations syzygies subschemes parametrized Hilbert scheme points $Hilb^d(Y)$, discuss applications to $Hilb^d(X)$ arbitrary intersections $X \mathbb{P}^n$.

Traditional formulations of geometric problems from the Schubert calculus, either in Plücker coordinates or in local coordinates provided by Schubert cells, yield systems of polynomials that are typically far from complete intersections and (in local coordinates) typically of degree exceeding two. We present an alternative primal-dual formulation using parametrizations of Schubert cells in the ...

We develop a formula (Theorem 5.1) which allows to compute top Chern classes of vector bundles on the vanishing locus V (s) of a section of this bundle. This formula particularly applies in the case when V (s) is the union of locally complete intersections giving the individual contribution of each component and their mutual intersections. We conclude with applications to the enumeration of rat...

We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.

We determine the simplicial complexes ∆ whose Stanley-Reisner ideals I∆ have the following property: for all n ≥ 1 the powers In ∆ have linear resolutions and finite length local cohomologies.

We characterize complete intersections in terms of local differential geometry. Let X ⊂ CPn+a be a variety. We first localize the problem; we give a criterion for X to be a complete intersection that is testable at any smooth point of X. We rephrase the criterion in the language of projective differential geometry and derive a sufficient condition for X to be a complete intersection that is com...

It has been shown by Batyrev and Borisov that nef partitions of reflexive polyhedra can be used to construct mirror pairs of complete intersection Calabi–Yau manifolds in toric ambient spaces. We construct a number of such spaces and compute their cohomological data. We also discuss the relation of our results to complete intersections in weighted projective spaces and try to recover them as sp...

The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this e...