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

Title: $c(\mathrm{Pol}^d(\mathbb{C}^n))$的斯特林系数和法诺纲数的多项式性

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. Show Chinese abstract

Abstract: 我们证明了$d$依赖于$c(\mathrm{Pol}^d(\mathbb{C}^n))$，即次数为$d$的$\mathrm{GL}(n)$表示的$n$个复变量的齐次多项式的陈类是多项式的。 我们也研究了这个陈类的多项式$d$依赖性的渐进行为。 我们将这些结果应用于解决Manivel关于包含线性子空间的超曲面簇的次数的猜想。 我们还证明了对于一般超曲面的Fano直线纲领的次数和欧拉特征的新公式。 为了证明$c(\mathrm{Pol}^d(\mathbb{C}^n))$的多项式依赖性，我们引入了斯特林系数的概念。 这可以看作是斯特林数的一个推广。 我们将斯特林系数表示为单项对称多项式的特化形式，因此我们可以推导出它们的多项式性质，并在某些情况下推导出它们的渐进行为。