Newton Binomial Formulas in Schubert Calculus

Abstract

We prove Newton's binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points.Key words: Schubert Calculus on a Grassmann algebra, Newton's binomial formulas in Schubert calculus, enumerative geometry of linear series on the projective line.

Equivariant Schubert calculus

“…Each λ∈ n k can also be represented by a string λ(1)λ (2)...λ(n) of zeros and ones only (as in [9]). If λ=a i1...i k , write S λ :=G μ (i1,...,i k ) (D)∈A * ( k M (p)) and…”

“…It turns out that H * T (G(k, n)), the integral T -equivariant cohomology ring of G(k, n), is a finite free module over the ring A:=H * T (pt), the T -equivariant cohomology of a point. Our main result (Theorem 2.2) shows that the multiplicative structure of the A-algebra H * T (G(k, n)), for all 1≤k≤n, can be described through derivations on the Grassmann algebra of a free A-module of rank n, in the same spirit as [2], [5], [6] and [7]. We stress that the description of the product structure of H * T (G(k, n)) heavily depends on the particular choice of an Abasis for it.…”

Tango bundles on Grassmannians

“…In the last part of this section, we will recall the basic definitions and results on Schubert calculus on the Grassmannian, paying special attention to the point of view of the subject given in and . Definition Let us fix a non empty flag of made of linear subspaces .…”

“…We will now recall a different point of view of Schubert calculus, using derivations, introduced in and by Gatto.…”

“…One of the key points is precisely the vanishing of this Chern class. In spite that the intersection product on is well‐known since long time ago, to get our result we must use a new point of view of Schubert calculus introduced recently in and and based on derivations.…”