Enumerative geometry of degeneracy loci

Abstract

Enumerative geometry of degeneracy lociAnnales scientifiques de l'É.N.S. 4 e série, tome 21, n o 3 (1988), p. 413-454 <http://www.numdam.org/item?id=ASENS_1988_4_21_3_413_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1988, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

The algebraic degree of semidefinite programming

“…In Section 5 we present results involving projective duality and determinantal varieties. in Section 6 this is combined with results of Pragacz [19] to prove Theorem 11, and to derive the general formula stated in Theorem 19 and Conjecture 21.…”

Common subbundles and intersections of divisors

“…It seems likely that the injectivity of π * could be extracted from work of Pragacz [9,Section 3]. He works with Chow groups of varieties rather than generalised cohomology rings of spaces, and his methods and language are rather different; we have not attempted a detailed comparison.…”

“…This creates a link with the theory of degeneracy loci and the corresponding classes in the cohomology of manifolds or Chow rings of varieties, which are given by the determinantal formula of Thom and Porteous. The paper [9] by Pragacz is a convenient reference for comparison with the present work. The relevant theory is based strongly on Schubert calculus, and could presumably be transferred to complex cobordism (and thus to other complex-orientable theories) by the methods of Bressler and Evens [1].…”

Grothendieck classes of quiver varieties

“…In this section we prove of a K-theory parallel of a Gysin formula of Pragacz [21,13]. We start with a Lemma which indicates that the classes G k (F ) are the right K-theoretic generalizations of Segre classes of a vector bundle F .…”

“…The class of Ω r can then be calculated inductively as the pushforward of the class of Ωr, which is done using a Gysin formula of Pragacz [21]. However, before this Gysin formula can be applied, one must first rearrange the inductive formula for Ωr by replacing the Schur polynomials s µi (E i − E i−1 ) in this formula with linear combinations of products s σ (E i − F ) · s τ (F − E i−1 ) for other bundles F , which can be done by invoking the coproduct in the ring of symmetric functions.…”