We introduce a new class of matroids, called graph curve matroids. A graph curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of hyperplane sections of degenerate canonical curves in algebraic geometry. Our focus lies on graphs that are 2-connected and trivalent, which define identically self-dual graph curve matroids, but we also develop generalizations. Finally, we provide an algorithm to compute the graph curve matroid associated to a given graph, as well as an implementation and data of examples that can be used in Macaulay2.---

Barnette's conjecture states that every cubic, bipartite, planar and 3-connected graph is Hamiltonian. Goodey verified Barnette's conjecture for all graphs with faces of size up to 6. We substantially strengthen Goodey's result by proving Hamiltonicity for cubic, bipartite, planar and (2-)connected graphs with faces of size up to 8. Parts of the proof are computational, including a distinction of 339.068.624 cases.

Volume polynomials measure the growth of Minkowski sums of convex bodies and of tensor powers of positive line bundles on projective varieties. We show that Aluffi's covolume polynomials are precisely the polynomial differential operators that preserve volume polynomials, reflecting a duality between homology and cohomology. We then present several applications to matroid theory.

In ???, a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties was introduced which we thus call ???. In this note we study thesearrangements and investigate their properties systematically. Remarkably the arrangements ??? discriminate between essentially all local properties of arrangements. In addition we show that hyperpolygonal arrangements are projectively unique and combinatorially formal. We note that the arrangement ??? is the famous counterexample of Edelman and Reiner ??? of Orlik's conjecture that the restriction of a free arrangement is again free.

Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a kth dilation is a sum of k lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the \(h^*\)-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that \(h^*\)-vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.

In this article we investigate the property of complete monotonicity within a special family Fs of functions in s variables involving logarithms. The main result of this work provides a linear isomorphism between Fs and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in Fs is semi-algebraic. This gives a finite time algorithm to decide whether a function in Fs is completely monotone.

We introduce a new multiplication for the polytope algebra, defined via the intersection of polytopes. After establishing the foundational properties of this intersection product, we investigate finite-dimensional subalgebras that arise naturally from this construction. These subalgebras can be regarded as volumetric analogues of the graded Möbius algebra, which appears in the context of the Dowling-Wilson conjecture. We conjecture that they also satisfy the injective hard Lefschetz property and the Hodge-Riemann relations, and we prove these in degree one.

This paper deals with sequences of random variables \(X_n\) only taking values in \(\{0,\ldots,n\}\). The probability generating functions of such random variables are polynomials of degree \(n\). Under the assumption that the roots of these polynomials are either all real or all lie on the unit circle in the complex plane, a quantitative normal approximation bound for \(X_n\) is established in a unified way. In the real rooted case the result is classical and only involves the variances of \(X_n\), while in the cyclotomic case the fourth cumulants or moments of \(X_n\) appear in addition. The proofs are elementary and based on the Stein--Tikhomirov method.

An arrangement of hypersurfaces in projective space is SNC if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomogeneous prime ideal of this variety.

Recently, Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph G, so called connected subgraph arrangements AG. In this note we strengthen some of the result from their work and prove new ones for members of this class. For instance, we show that aspherical members withing this class stem from a rather restricted set of graphs. Specifically, if AG is an aspherical connected subgraph arrangement, then AG is free with the unique possible exception when the underlying graph G is the complete graph on 4 nodes.