发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究拟阵的Snapper多项式的魔幻正性，通过引入zonotopal类和饱和K-类，证明其系数正性及h*-多项式实根性，并推广到模空间上的线丛。
AI 中文摘要
1959年，Snapper证明了正规射影概形上一条线丛的张量幂的欧拉示性数是一个多项式，后来被称为Snapper多项式。Snapper多项式系数的正性蕴含了线丛的各种正性概念，我们通过魔幻正性和实根性来研究这些概念。对于任何无环拟阵，我们在环面簇的向量丛的Grothendieck $K$-环中引入zonotopal类，并证明它们的Snapper多项式是魔幻正的。我们的证明将这样的Snapper多项式实现为拟阵沿某些直线的Dilworth截断的加权独立多项式。作为推论，它们的系数是正的，并且它们的$h^{\ast}$-多项式是实根的。在可实现情形下，该多项式是嵌入在射影直线乘积中的奇妙簇的多重分次Hilbert多项式。我们在Deligne--Mumford--Knudsen模空间$\overline{\mathcal M}_{0,n}$上引入类似的线丛，并证明它们的Snapper多项式是魔幻正的。对于第一陈类为不同的$\psi$-类的余切丛线丛（它们不是zonotopal的），我们仍然证明它们的$h^{\ast}$-多项式是实根的，而它们的Snapper多项式是魔幻正的当且仅当$n\leqslant7$。更一般地，我们引入拟阵的饱和和弱饱和$K$-类，它们以龙Hall--Rado多拟阵的形式为Snapper多项式的魔幻正性提供了充分条件和必要条件。
英文摘要
In 1959, Snapper showed that the Euler characteristic of the tensor powers of a line bundle on a normal projective scheme is a polynomial, later named the \emph{Snapper polynomial}. Positivity of coefficients of Snapper polynomials implies various notions of positivity of line bundles, which we study through the lens of magic positivity and real-rootedness. We introduce zonotopal classes in the Grothendieck $K$-ring of vector bundles of the toric variety for any loopless matroid, and prove that their Snapper polynomials are magic positive. Our proof realizes such a Snapper polynomial as a weighted independence polynomial of the Dilworth truncation along certain lines of the matroid. As a consequence, their coefficients are positive, and their $h^{\ast}$-polynomials are real-rooted. In the realizable case, this polynomial is the multigraded Hilbert polynomial of the wonderful variety embedded in a product of projective lines. We introduce analogous line bundles on the Deligne--Mumford--Knudsen moduli space $\overline{\mathcal M}_{0,n}$ and prove that their Snapper polynomials are magic positive. For cotangent line bundles whose first Chern classes are distinct $ψ$-classes, which are not zonotopal, we nonetheless prove that their $h^{\ast}$-polynomials are real-rooted, whereas their Snapper polynomials are magic positive if and only if $n\leqslant7$. More generally, we introduce saturated and weakly saturated $K$-classes of matroids, which furnish a sufficient and a necessary condition for magic positivity of Snapper polynomials in terms of their dragon Hall--Rado polymatroids.
Comments32 pages; comments welcome!
