Conormal Spaces and Whitney Stratifications

Abstract

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps.

Correction to: Conormal Spaces and Whitney Stratifications

“…Let X be an algebraic variety in R n . The WhitStrat algorithm of [14], when applied to X(C), produces a Whitney stratification of X.…”

“…From real to complex and back. In prior work, we used conormal spaces and primary decomposition to algorithmically stratify complex algebraic varieties [14]. In the introductory remarks to that paper, we highlighted the lack of Gröbner basis techniques over R as a primary obstacle to performing similar stratifications for real algebraic varieties and semialgebraic sets.…”

“…The key insight is that the subvarieties arising from a stratification of X(C) produced by the methods of [14] are also generated by real polynomials; and the real varieties defined by those polynomials constitute a valid Whitney stratification of X.…”

“…Whenever f is proper (i.e., if X is compact) then the restriction of f to f −1 (N) forms a locally trivial fiber bundle over N, which in particular implies that the stratified homeomorphism type of the fiber f −1 (y) is independent of the choice of y ∈ N. Our second contribution, carried out in Section 3, is to describe an algorithm for stratifying any given f . As with real varieties, the key step is to first consider a complexified version f C : X(C) → Y(C) of the morphism, and to then employ the methods of [14].…”