Small Subalgebras of Polynomial Rings and Stillman's Conjecture

Abstract

Let n, d, η be positive integers. We show that in a polynomial ring R in N variables over an algebraically closed field K of arbitrary characteristic, any K-subalgebra of R generated over K by at most n forms of degree at most d is contained in a K-subalgebra of R generated by B ≤ η B(n, d) forms G 1 , . . . , G B of degree ≤ d, where η B(n, d) does not depend on N or K, such that these forms are a regular sequence and such that for any ideal J generated by forms that are in the K-span of G 1 , . . . , G B , the ring R/J satisfies the Serre condition Rη. These results imply a conjecture of M. Stillman asserting that the projective dimension of an n-generator ideal of R whose generators are forms of degree ≤ d is bounded independent of N . We also show that there is a primary decomposition of the ideal such that all numerical invariants of the decomposition (e.g., the number of primary components and the degrees and numbers of generators of all of the prime and primary ideals occurring) are bounded independent of N .

On the codimension of the singular locus

“…The following statement is proven in [1]. The goal of this paper is to obtain an explicit upper bound on C(d, c, m).…”

On Bias and Rank

“…In this section we shall connect c-regularity to notion of rank, one key result we shall use is Theorem A. [AH16]. It should be noted though that in [AH16] our notion of rank is called strength.…”

“…[AH16]. It should be noted though that in [AH16] our notion of rank is called strength. Now given F a polynomial of degree d over F q .…”

Linear subspaces of hypersurfaces

“…To our knowledge, this is the first result on the de Jong-Debarre Conjecture that works for all degrees d and does not require n to grow exponentially with d. The technique relies on a new result that says, essentially, smooth high degree hypersurfaces tend not to be tangent to varieties cut out by lower-degree equations (see Lemma 2.1). The approach is somewhat similar in philosophy, although not in technique, to results of Ananyan, Hochster, Erman, Sam, and Snowden [1,9] in that it describes how a fixed number of smooth equations tend to become algebraically independent as the number of variables grows.…”