Magic positivity of Snapper polynomials for matroids

Source abstract

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.