arXiv subjects

Alessio Caminata

Publications and source records attributed to Alessio Caminata.

28 records · Page 2Linked to original sources

arXiv2021

Point configurations, phylogenetic trees, and dissimilarity vectors

In 2004 Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter--Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

math.AG↗

arXiv2020

Nearly Gorenstein cyclic quotient singularities

We investigate the nearly Gorenstein property among $d$-dimensional cyclic quotient singularities $\Bbbk[[x_1,\dots,x_d]]^G$, where $\Bbbk$ is an algebraically closed field and $G\subseteq{\rm GL}(d,\Bbbk)$ is a finite small cyclic group whose order is invertible in $\Bbbk$. We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.

math.AC↗

arXiv2019

Equations for point configurations to lie on a rational normal curve

The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the birational geometry of the moduli space $\overline{\mathrm{M}}_{0,n}$ of $n$-tuples of points in $\mathbb{P}^1$. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely $d=2$, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, $d=3$, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For $d \ge 4$ we conjecture a similar situation and prove partial results in this direction.

math.AG↗

arXiv2018

Differential symmetric signature in high dimension

We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings $k[x_1,\dots,x_n]^G$ where $G$ is a finite small subgroup of $\mathrm{Gl}(n,k)$ and for hypersurface rings $k[x_1,\dots,x_n]/(f)$ of dimension $\geq3$ with an isolated singularity. In the first case, we obtain the value $1/|G|$, which coincides with the F-signature and generalizes a previous result of the authors for the two-dimensional case. In the second case, following an argument by Bruns, we obtain the value $0$, providing an example of a ring where differential symmetric signature and F-signature are different.

math.AC↗

arXiv2018

F-signature function of quotient singularities

We study the shape of the F-signature function of a $d$-dimensional quotient singularity $\mathbb{k}[[ x_1,\ldots,x_d]]^G$, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group $G\subseteq \mathrm{Gl}(d,\Bbbk)$. When $G$ is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples.

math.AC↗

arXiv2016

The symmetric signature

We define two related invariants for a $d$-dimensional local ring $(R,\mathfrak{m},k)$ called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two modules: the top dimensional syzygy module $\mathrm{Syz}^d_R(k)$ of the residue field and the module of Kähler differentials $Ω_{R/k}$ of $R$ over $k$. We compute these invariants for two-dimensional ADE singularities obtaining $1/|G|$, where $|G|$ is the order of the acting group, and for cones over elliptic curves obtaining $0$ for the differential symmetric signature. These values coincide with the F-signature of such rings in positive characteristic.

math.AC↗

arXiv2016

Generalized Hilbert-Kunz function in graded dimension two

We prove that the generalized Hilbert-Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+γ(q)$, with rational generalized Hilbert-Kunz multiplicity $e_{gHK}(M)$ and a bounded function $γ(q)$. Moreover we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\rightarrow+\infty$ of the generalized Hilbert-Kunz multiplicity $e_{gHK}^{R_p}(M_p)$ over the fibers $R_p$ exists and it is a rational number.

math.AC↗

arXiv2016

The symmetric signature

This is the author's Ph.D. thesis. We introduce two related invariants for local (and standard graded) rings called differential and syzygy symmetric signature. These are defined by looking at the maximal free splitting of the module of Kähler differentials and of the the top-dimensional syzygy module of the residue field respectively. We study and compute them for different classes of rings: two-dimensional ADE singularities, two-dimensional cyclic singularities, and cones over plane elliptic curves (for the differential symmetric signature). The values obtained coincide with the F-signature of such rings in positive characteristic. The thesis contains also a short survey on the Auslander correspondence for quotient singularities.

math.AC↗

arXiv2016

The Symmetric signature of cyclic quotient singularities

The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e. invariant subrings $k[[u,v]]^G$ of rings of formal power series under the action of a cyclic group $G$. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincides with the $F$-signature.

math.AC↗

arXiv2015

Cohomological dimension and arithmetical rank of some determinantal ideals

Let $M$ be a $(2 \times n)$ non-generic matrix of linear forms in a polynomial ring. For large classes of such matrices, we compute the cohomological dimension (cd) and the arithmetical rank (ara) of the ideal $I_2(M)$ generated by the $2$-minors of $M$. Over an algebraically closed field, any $(2 \times n)$-matrix of linear forms can be written in the Kronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and nilpotent blocks. Bădescu and Valla computed $\mathrm{ara}(I_2(M))$ when $M$ is a concatenation of scroll blocks. In this case we compute $\mathrm{cd}(I_2(M))$ and extend these results to concatenations of Jordan blocks. Eventually we compute $\mathrm{ara}(I_2(M))$ and $\mathrm{cd}(I_2(M))$ in an interesting mixed case, when $M$ contains both Jordan and scroll blocks. In all cases we show that $\mathrm{ara}(I_2(M))$ is less than the arithmetical rank of the determinantal ideal of a generic matrix.

math.AC↗