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Lorentzian polynomials, Segre classes, and adjoint polynomials of convex polyhedral cones by Aluffi, Paolo is a scholarly article available to read on EtoBox.

What is Lorentzian polynomials, Segre classes, and adjoint polynomials of convex polyhedral cones about?

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.

Author

Aluffi, Paolo

Published

2023

Language

EN