tailieunhanh - Đề tài " Schubert induction "

We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule of [V2]. As applications, we show that all Schubert problems for all Grassmannians are enumerative over the real numbers, and sufficiently large finite fields. We prove a generic smoothness theorem as a substitute for the Kleiman-Bertini theorem in positive characteristic. We compute the monodromy groups of many Schubert problems, and give some surprising examples where the monodromy group is much smaller than the full symmetric group. . | Annals of Mathematics Sch ubert induction By Ravi Vakil Annals of Mathematics 164 2006 489 512 Schubert induction By Ravi Vakil Abstract We describe a Schubert induction theorem a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule of V2 . As applications we show that all Schubert problems for all Grassmannians are enumerative over the real numbers and sufficiently large finite fields. We prove a generic smoothness theorem as a substitute for the Kleiman-Bertini theorem in positive characteristic. We compute the monodromy groups of many Schubert problems and give some surprising examples where the mon-odromy group is much smaller than the full symmetric group. Contents 1. Questions and answers 2. The main theorem and its proof 3. Galois monodromy groups of Schubert problems References The main theorem of this paper Theorem is an inductive method Schubert induction of proving results about intersections of Schubert varieties in the Grassmannian. In Section 1 we describe the questions we wish to address. The main theorem is stated and proved in Section 2 and applications are given there and in Section 3. 1. Questions and answers Fix a Grassmannian G k n G k 1 n 1 over a base field or ring K. Given a partition a the condition of requiring a k-plane V to satisfy dim V n Fn-ai i i with respect to a flag F is called a Schubert condition. The Partially supported by NSF Grant DMS-0228011 an AMS Centennial Fellowship and an Alfred P. Sloan Research Fellowship. 490 RAVI VAKIL variety of fc-planes satisfying a Schubert condition with respect to a flag F is the Schubert variety Qa F . Let Qa G A G k n denote the corresponding Schubert class. Let Qa F c G k n X Fl n be the universal Schubert variety . A Schubert problem is the following Given m Schubert conditions QQi F with respect to fixed general flags F 1 i m whose total codimension is dim G k n what is the .