Kapranov degrees (with Joshua Brakensiek, Christopher Eur, Matt Larson)
The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of ψ-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding "Kapranov degrees," which include as special cases the work of Witten, Silversmith, Castravet-Tevelev, Postnikov, Cavalieri-Gillespie-Monin, and Gillespie-Griffins-Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes.
Equivariant log-concavity of independence sequences of claw-free graphs Séminaire Lotharingien de Combinatoire 89B.
We show that the graded vector space spanned by independent vertex sets of any claw-free graph is strongly equivariantly log-concave, viewed as a graded permutation representation of the graph automorphism group. Our proof reduces the problem to the equivariant hard Lefschetz theorem on the cohomology of a product of projective lines. Both the result and the proof generalize our previous result on graph matchings. This also gives a strengthening and a new proof of results of Hamidoune, and Chudnovsky--Seymour.
K-rings of wonderful varieties and matroids (with Matt Larson, Sam Payne, Nick Proudfoot)
We study the K-ring of the wonderful variety of a hyperplane arrangement and give a combinatorial presentation that depends only on the underlying matroid. We give a new invariant of arbitrary matorids. It computes the Euler characteristics of arbitrary line bundles on wonderful varieties and on Deligne--Mumford--Knudsen moduli spaces of stable rational curves with marked points.
Intersecting psi-classes on tropical Hassett spaces (with Marvin Anas Hahn)
Combinatorial Theory, 2(3). (2022).
We study the intersection of tropical psi-classes on tropical heavy/light Hassett spaces, generalising a result of Kerber--Markwig. Our computation reveals that the weight of a maximal cone in an intersection has a combinatorial intepretation in terms of the underlying tropical curve and it is always nonnegative. In particular, our result specialises to that, in top dimension, the tropical intersection product coincides with its classical counterpart.
Relative Bott--Samelson varieties La Matematica 2. (2023)
We generalize Brion's Bott--Samelson resolution of singularities of Richardson varieties to a relative setting. We then applied it to give a resolution of singularities for the Brill-Noether variety with imposed ramification on twice-marked elliptic curves, studied by Chan--Osserman--Pflueger.