Polynomiality of the Striling coefficients of $c(\mathrm{Pol}^d(\mathbb{C}^n))$ and Fano schemes

Abstract

We prove that the $d$-dependence of $c(\mathrm{Pol}^d(\mathbb{C}^n))$, the Chern class of the $\mathrm{GL}(n)$-representation of degree $d$ homogeneous polynomials in $n$ complex variables is polynomial. We also study the asymptotics of the polynomial $d$-dependence of this Chern class. We apply these results to solve a conjecture of Manivel on the degree of varieties of hypersurfaces containing linear subspaces. We also prove new formulas for the degree and Euler characteristics of Fano schemes of lines for generic hypersurfaces. To prove the polynomial dependence of $c(\mathrm{Pol}^d(\mathbb{C}^n))$ we introduce the notion of Striling coefficients. This can be considered as a generalization of Stirling numbers. We express the Stirling coefficients in terms of specializations of monomial symmetric polynomials so we can deduce their polynomiality and, in certain cases, their asymptotic behaviour.