Lorentzian polynomials, Segre classes, and adjoint polynomials of convex polyhedral cones
Paolo Aluffi
TL;DR
This work introduces covolume polynomials as a dual analogue to volume polynomials and analyzes their relation to Lorentzian polynomials. It establishes sectional log-concavity and $M$-convex support for covolume polynomials and shows that Chern classes of globally generated bundles yield covolume polynomials; it proves that numerators of Segre zeta functions of subschemes in products of projective spaces are covolume polynomials under natural hypotheses, with univariate coefficients exhibiting log-concavity. It further proves that adjoint polynomials of convex polyhedral cones in the nonnegative orthant are covolume polynomials, yielding $M$-convexity and sectional log-concavity and, after a change of variables, dual Lorentzian behavior. The results unify intersection-theoretic invariants with polyhedral geometry, suggesting new structural insights and conjectures about Lorentzian normalization in broader settings.
Abstract
We consider polynomials expressing the cohomology classes of subvarieties of products of projective spaces, and limits of positive real multiples of such polynomials. We study the relation between these covolume polynomials and Lorentzian polynomials. While these are distinct notions, we prove that, like Lorentzian polynomials, covolume polynomials have M-convex support and generalize the notion of log-concave sequences. In fact, we prove that covolume polynomials are `sectional log-concave', that is, the coefficients of suitable restrictions of these polynomials form log-concave sequences. We observe that Chern classes of globally generated bundles give rise to covolume polynomials, and use this fact to prove that certain polynomials associated with Segre classes of subschemes of products of projective spaces are covolume polynomials. We conjecture that the same polynomials may be Lorentzian after a standard normalization operation. Finally, we obtain a combinatorial application of a particular case of our Segre class result. We prove that the adjoint polynomial of a convex polyhedral cone contained in the nonnegative orthant, and sharing a face with it, is a covolume polynomial. This implies that these adjoint polynomials are M-convex and sectional log-concave, and in fact dually Lorentzian, that is, Lorentzian after a certain change of variables.