Simplicial complexes in Macaulay2
We highlight some features of the SimplicialComplexes package in Macaulay2.
math.AC↗
arXiv subjects
Alexandre Zotine
Publications and source records attributed to Alexandre Zotine.
Numerical computation of endomorphism rings
We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties.
math.NT↗
On Fano Schemes of Toric Varieties
Let $X_\mathcal{A}$ be the projective toric variety corresponding to a finite set of lattice points $\mathcal{A}$. We show that irreducible components of the Fano scheme $\mathbf{F}_k(X_\mathcal{A})$ parametrizing $k$-dimensional linear subspaces of $X_\mathcal{A}$ are in bijection to so-called maximal Cayley structures for $\mathcal{A}$. We explicitly describe these irreducible components and their intersection behaviour, characterize when $\mathbf{F}_k(X_\mathcal{A})$ is connected, and prove that if $X_\mathcal{A}$ is smooth in dimension $k$, then every component of $\mathbf{F}_k(X_\mathcal{A})$ is smooth in its reduced structure. Furthermore, in the special case $k=\dim X_\mathcal{A}-1$, we describe the non-reduced structure of $\mathbf{F}_k(X_\mathcal{A})$. Our main result is closely related to concurrent work done independently by Furukawa and Ito.
math.AG↗