Union Of Strata Research Articles

Closure relations of Newton strata in Iwahori double cosets

We consider the Newton stratification on Iwahori double cosets for a connected reductive group. We prove the existence of Newton strata whose closures cannot be expressed as a union of strata, and show how this is implied by the existence of non-equidimensional affine Deligne–Lusztig varieties. We also give an explicit example for a group of type A_4.

Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations: 33 Years Later. I

Consider a system of polynomial equations with parametric coefficients over an arbitrary ground field. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the solutions of the system are given by algebraic formulas depending only on this stratum. Each stratum is a quasiprojective algebraic variety with degree bounded from above by a subexponential function in the size of the input data. The number of strata is also subexponential in the size of the input data. Thus, here we avoid double exponential upper bounds on the degrees and solve a long-standing problem

Efficient Absolute Factorization of Polynomials with Parametric Coefficients

Consider a polynomial with parametric coefficients. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the decomposition of this polynomial into absolutely irreducible factors is given by algebraic formulas depending only on the stratum. Each stratum is a quasiprojective algebraic variety. This variety and the corresponding output are given by polynomials of degrees at most D with D = d′d O(1) where d′, d are bounds on the degrees of the input polynomials. The number of strata is polynomial in the size of the input data. Thus, here we avoid double exponential upper bounds for the degrees and solve a long-standing problem.

Explicit Models for Perverse Sheaves, II

In this paper we show that any p-perverse sheaf on an arbitrary stratified topological space (p is a perversity function) is functorially determined by a system of usual sheaves on the open sets U r (r≥0) and certain gluing data, where U r is the union of strata of perversity ≤r.

Geometry of Hilbert modular varieties over totally ramified primes

We study mod p Hilbert modular varieties associated to a totally real field L of degree g under the assumption that p is maximally ramified in OL, the ring of algebraic integers of L. We associate to each OL-abelian variety A over an Fp-algebra integer invariants 0 ≤ j ≤ n ≤ g−j that are used to define disjoint locally closed sets W(j,n) of the moduli space. The invariants encode some of the structure of H1(A, OA) as an OL[F]-module. We prove that the sets W(j,n) are nonempty, regular subvarieties of dimension g−(j+n). Moreover, the Newton polygon is constant on W(j,n) having two slopes (λ(n), 1−λ(n)), each with multiplicity g, where λ(n) = min(1/2, n/g). Using Hecke correspondences at p and Moret-Bailly families, we prove that the collection {W(j,n): 0 ≤ j ≤ n ≤ g−j} is indeed a stratification: the boundary is a union of strata. Furthermore, the singularity stratification of Deligne-Pappas, the Newton stratification, and the a-number stratification, all have a simple expression as a union of our strata. We affirm general conjectures on Newton strata for this class of varieties, that they form a stratification, that they have the expected dimension (each strata being of codimension 1 in the previous strata), and that the Grothendieck conjecture holds.