arXiv subjects

Laura Casabella

Publications and source records attributed to Laura Casabella.

7 recordsLinked to original sources

arXiv2025

The variety of orthogonal frames

An orthogonal n-frame is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames in a d-dimensional quadratic vector space is an algebraic variety V(d,n). In this paper, we investigate the variety V(d,n) as well as the quadratic ideal I(d,n) generated by the orthogonality relations, which cuts out V(d,n). We classify the irreducible components of V(d,n), give criteria for the ideal I(d,n) to be prime or a complete intersection, and for the variety V(d,n) to be normal. We also give near-equivalent conditions for V(d,n) to be factorial. Applications are given to the theory of Lovász-Saks-Schrijver ideals.

math.AC↗

arXiv2025

Tropical elliptic curves in 3-space

We classify trivalent graphs with 16 vertices and 16 edges that arise from intersecting two quadratic surfaces in tropical 3-space. There are 4,009 such graphs, representing maximally degenerate stable models of elliptic curves realized as tropical complete intersections of two quadrics. Our classification is derived from 405,246,030 regular unimodular triangulations of the 4-dimensional Cayley polytope.

math.CO↗

arXiv2024

Wronski Pairs of Honeycomb Curves

We study certain generic systems of real polynomial equations associated with triangulations of convex polytopes and investigate their number of real solutions. Our main focus is set on pairs of plane algebraic curves which form a so-called Wronski system. The computational tasks arising in the analysis of such Wronski pairs lead us to the frontiers of current computer algebra algorithms and their implementations, both via Gröbner bases and numerical algebraic geometry.

math.AG↗

arXiv2024

Subdivisions of Hypersimplices: with a View Toward Finite Metric Spaces

The secondary fan $Σ(k,n)$ is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices $Δ(k,n)$. We find new infinite families of rays of $Σ(k,n)$, and we compute the fans $Σ(2,7)$ and $Σ(3,6)$. In the special case $k=2$ the fan $Σ(2,n)$ is closely related to the metric fan $\mathop{MF}(n)$, which forms a natural parameter space for the metric spaces on $n$ points. So our results yield a classification of the finite metric spaces on seven points.

math.CO↗

arXiv2023

Standard monomial theory modulo Frobenius in characteristic two

Over a field of characteristic two, we develop a theory of standard monomials for polynomial rings modulo a Frobenius power of the maximal ideal generated by all variables. As a result, we obtain a filtration by modular GL_n-representations whose characters are given by particular truncated Schur polynomials, thus proving a conjecture by Gao--Raicu--VandeBogert in the characteristic two case.

math.AC↗

arXiv2023

Apéry sets and the ideal class monoid of a numerical semigroup

The aim of this article is to study the ideal class monoid $\mathcal{C}\ell(S)$ of a numerical semigroup $S$ introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $\mathcal{C}\ell(S)$. We observe that $\mathcal{C}\ell(S)$ is isomorphic to the monoid of ideals of $S$ whose smallest element is 0, which helps to relate $\mathcal{C}\ell(S)$ to the Apéry sets and the Kunz coordinates of $S$. We study some combinatorial and algebraic properties of $\mathcal{C}\ell(S)$, including the reduction number of ideals, and the Hasse diagrams of $\mathcal{C}\ell(S)$ with respect to inclusion and addition. From these diagrams we can recover some notable invariants of the semigroup. Lastly, we prove some results about irreducible elements, atoms, quarks and primes of $(\mathcal{C}\ell(S),+)$. Idempotent ideals coincide with over-semigroups and idempotent quarks correspond to unitary extensions of the semigroup. We show that a numerical semigroup is irreducible if and only if $\mathcal{C}\ell(S)$ has at most two quarks.

math.AC↗

arXiv2022

Almost symmetric good semigroups

The class of good semigroups is a class of subsemigroups of $N^h$, that includes the value semigroups of rings associated to curve singularities and their blowups, and allows to study combinatorically the properties of these rings. In this paper we give a characterization of almost symmetric good subsemigroups of $N^h$, extending known results in numerical semigroup theory and in one-dimensional ring theory, and we apply these results to obtain new results on almost Gorenstein one-dimensional analytically unramified rings.

math.AC↗