×

Schubert induction. (English) Zbl 1115.14043

Consider the Grassmannian \(\mathcal G_K(k,n)\) over any field \(K\), some flags \(F_i\), \(i=1,\dots,m\), and consider the corresponding sets of conditions \(\Omega(F_i)\) on the intersection of \(k\)-planes with the elements of each \(F_i\). This datum defines a Schubert problem \(\omega=\{\Omega(F_i)\}\). The author studies the case where the expected set of solutions is finite. The problem \(\omega\) is enumerative over \(K\) when one finds the expected number \(\deg(\{\Omega(F_i)\})\) of solutions, formed by distinct \(k\)-planes, all of them defined over \(K\).
The author proves that any Schubert problem is enumerative over \(\mathbb R\), as well as over algebraically closed field of any characteristic. Moreover, he finds that Schubert problems over finite fields are enumerative in a set of positive density. One tool for proving the results is an extension of the Bertini-Kleiman smoothness condition. The author proves that the map from the set of solutions in the universal Grassmannian, to the product of \(m\) copies of the flag variety, is generically smooth.
The author finally studies the Galois group of an enumerative problem, for the Grassmannians of lines and planes.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14N15 Classical problems, Schubert calculus
14N10 Enumerative problems (combinatorial problems) in algebraic geometry

Keywords:

Schubert cycles
PDFBibTeX XMLCite
Full Text: DOI arXiv