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Antonino Ficarra

Publications and source records attributed to Antonino Ficarra.

At least 19 recordsLinked to original sources
arXiv2026

Symbolic Rees algebras of complementary edge ideals

Let $G$ be a finite simple graph on $[n]$ and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring $S = K[x_1,\dots,x_n]$. We give a combinatorial description, in terms of the structure of $G$, of the minimal generators of the symbolic Rees algebra $\mathcal{R}_s(I_c(G)) = \bigoplus_{k \geq 0} I_c(G)^{(k)} t^k$, and show that this algebra is generated in degree at most $6$. Moreover, we completely determine the minimal generators of $\mathcal{R}_{s}(I_{c}(G))$ in graph-theoretic terms. We then study in more detail the homological invariants of the symbolic powers $I_c(G)^{(k)}$ for the classes of cycle graphs and complete multipartite graphs. For theses families, we study the behavior of the symbolic depth function $k\mapsto\operatorname{depth} S/I_c(G)^{(k)}$, we obtain the limit depth of the symbolic powers and the Waldschmidt constant of $I_c(G)$, and further prove that all the symbolic powers $I_c(G)^{(k)}$ are componentwise linear.

math.AC↗
arXiv2026

Serre depth and local cohomology

We introduce a fundamental homological invariant, called Serre depth, which stratifies Serre's conditions in the same way that depth stratifies the Cohen-Macaulay property. We study the Serre depths of modules over arbitrary Noetherian local rings and over standard graded algebras over a field, extending the polynomial ring case due to Muta and Terai. Under mild hypotheses, we show that the $r$-th Serre depth of a finitely generated module $M$ measures the deviation of $M$ from satisfying Serre's condition $(S_r)$. The main results of the paper can be summarized as follows: (i) We establish the basic properties of Serre depth and prove that it is invariant under completion. (ii) If the base ring $R$ is a homomorphic image of a Gorenstein ring, we show that a finitely generated $R$-module $M$ is equidimensional and satisfies $(S_r)$ if and only if its $r$-th Serre depth equals its Krull dimension. Analogous statements are obtained for schemes. (iii) For a homogeneous ideal in a standard graded polynomial ring over a field, we compare its Serre depths with those of its initial ideal. (iv) We characterize the Serre depths of a monomial ideal in terms of its skeletons and prove that the Serre depths of sufficiently large powers of a monomial ideal stabilize; the proof uses Presburger arithmetic.

math.AC↗
arXiv2026

The stable set of associated primes of a complementary edge ideal

We explicitly determine the associated primes of every power of a complementary edge ideal, prove that they satisfy the persistence property, and compute the $\text{v}$-function. In the course of the proofs, we completely describe the homological properties of all powers of squarefree monomial ideals generated in degrees large relative to the number of variables defining them.

math.AC↗
arXiv2025

On the gradient of a monomial ideal

Let $K$ be a field of characteristic zero, let $I \subset S = K[x_1,\dots,x_n]$ be a homogeneous ideal, and let $\partial(I)$ be its gradient ideal. We study the relationship between $\mathrm{reg}\,I$ and $\mathrm{reg}\,\partial(I)$. While earlier work by Busé, Dimca, Schenck, and Sticlaru showed these regularities are generally incomparable for hypersurface ideals, we prove they remain incomparable even for monomial ideals with linear resolution, answering a question of J. Herzog. In fact, for any integers $a \in \mathbb{Z}$ and $b \ge - 1$, we construct monomial ideals $I$ and $J$ such that $\mathrm{reg}\,I - \mathrm{reg}\,\partial(I) = a$, $\mathrm{reg}\,\partial(J) - \mathrm{reg}\,J = b$ and $J$ has linear resolution. We introduce monomial ideals with differential linear resolution as those monomial ideals whose all iterated gradient ideals have linear resolution. We prove that polymatroidal ideals, equigenerated (strongly) stable ideals, powers of edge ideals with linear resolution, complementary edge ideals with linear resolution, and certain equigenerated squarefree monomial ideals with many generators satisfy this property.

math.AC↗
arXiv2025

Rees algebras of complementary edge ideals

In this paper we investigate the Rees algebras of squarefree monomial ideals $I \subset S=K[x_1,\dots,x_n]$ generated in degree $n-2$, where $K$ is a field. Every such ideal arises as the complementary edge ideal $I_c(G)$ of a finite simple graph $G$. We describe the defining equations of the Rees algebra $\mathcal{R}(I_c(G))$ in terms of the combinatorics of $G$. If $G$ is a tree or a unicyclic graph whose unique induced cycle has length $3$ or $4$, we prove that $\mathcal{R}(I_c(G))$ is Koszul. We also determine the asymptotic depth of the powers of $I_c(G)$, proving that $\lim_{k \to \infty}\text{depth}\, S/I_c(G)^k=b(G)$, where $b(G)$ is the number of bipartite connected components of $G$. Finally, we show that the index of depth stability of $I_c(G)$ is at most $n-2$, and equality holds when $G$ is a path graph.

math.AC↗
arXiv2025

Polymatroidal ideals and their asymptotic syzygies

Let $I$ be a polymatroidal ideal. In this paper, we study the asymptotic behavior of the homological shift ideals of powers of polymatroidal ideals. We prove that the first homological shift algebra $\text{HS}_1(\mathcal{R}(I))$ of $I$ is generated in degree one as a module over the Rees algebra $\mathcal{R}(I)$ of $I$. We conjecture that the $i$th homological shift algebra $\text{HS}_i(\mathcal{R}(I))$ of $I$ is generated in degrees $\le i$, and we confirm it in many significant cases. We show that $I$ has the $1$st homological strong persistence property, and we conjecture that the sequence $\{\text{Ass}\,\text{HS}_i(I^k)\}_{k>0}$ of associated primes of $\text{HS}_i(I^k)$ becomes an increasing chain for $k\ge i$. This conjecture is established when $i=1$ and for many families of polymatroidal ideals. Finally, we explore componentwise polymatroidal ideals, and we prove that $\text{HS}_1(I)$ is again componentwise polymatroidal, if $I$ is componentwise polymatroidal.

math.AC↗
arXiv2025

Complementary edge ideals

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as established in [11], in this work we investigate the algebraic and homological properties of $I$ and its powers. To this end, we introduce the complementary edge ideal of a finite simple graph $G$ as the ideal $$I_c(G)=((x_1\cdots x_n)/(x_ix_j):\{i,j\}\in E(G)) $$ of $S$, where $V(G)=\{1,\ldots,n\}$ and $E(G)$ is the edge set of $G$. By interpreting any squarefree monomial ideal $I$ generated in degree $n-2$ as the complementary edge ideal of a graph $G$, we establish a correspondence between algebraic invariants of $I$ and combinatorial properties of $G$. More precisely, we characterize sequentially Cohen-Macaulay, Cohen-Macaulay, Gorenstein, nearly Gorenstein and matroidal complementary edge ideals. Moreover, we determine the regularity of powers of $I$ in terms of combinatorial invariants of the graph $G$ and obtain that $I^k$ has linear resolution or linear quotients for some $k$ (equivalently for all $k\geq 1$) if and only if $G$ has only one connected component with at least two vertices.

math.AC↗
arXiv2025

Stanley-Reisner ideals with linear powers

Let $S = K[x_1, \dots, x_n]$ be the standard graded polynomial ring over a field $K$. In this paper, we address and completely solve two fundamental open questions in Commutative Algebra: (i) For which degrees $d$, does there exist a uniform combinatorial characterization of all squarefree monomial ideals in $S$ having $d$-linear resolutions? (ii) For which degrees $d$, does having a linear resolution coincide with having linear powers for all squarefree monomial ideals of $S$ generated in degree $d$? Let $\mathcal{I}_{n,d}(K)$ denote the class of squarefree monomial ideals of $S$ having a $d$-linear resolution. Our main result establishes the equivalence of the following conditions: (a) Any squarefree monomial ideal $I$ in $S$ generated in degree $d$ has a linear resolution, if and only if, $I$ has linear powers. (b) $\mathcal{I}_{n,d}(K)$ is independent of the base field $K$. (c) $d\in\{0,1,2,n{-}2,n{-}1,n\}$. In each of these degrees, we show that a squarefree monomial ideal has a linear resolution if and only if all of its powers admit linear quotients, and we combinatorially classify such ideals. In contrast, for each degree $3\le d\le n{-}3$, we construct fully-supported squarefree monomial ideals $I$ and $J$ in $S$ generated in degree $d$ such that the linear resolution property of $I$ depends on the choice of the base field, $J$ has a linear resolution and $J^2$ does not have a linear resolution.

math.AC↗
arXiv2025

Dao numbers and the asymptotic behaviour of fullness

In the present paper, we study the Dao numbers $\mathfrak{d}_1(I),\mathfrak{d}_2(I)$ and $\mathfrak{d}_3(I)$ of an ideal $I$ of a Noetherian local ring $(R,\mathfrak{m},K)$ or a standard graded Noetherian $K$-algebra. They are defined as the smallest $\ell\ge0$ such that $I\mathfrak{m}^k$ is $\mathfrak{m}$-full, full, weakly $\mathfrak{m}$-full, respectively, for all $k\ge\ell$. We provide general bounds for the Dao numbers in terms of the Castelnuovo-Mumford regularity of certain modules over the Rees algebra $\mathcal{R}(\mathfrak{m})$. If $R$ is a Koszul algebra, we prove that the Dao numbers are less or equal to $\text{reg}_{\text{gr}_\mathfrak{m}(R)}\text{gr}_\mathfrak{m}(I)$, where $\text{gr}_\mathfrak{m}(I)$ is the associated graded module of $I$. Finally, for monomial ideals, we combinatorially bound the Dao numbers in terms of asymptotic linear quotients and bounding multidegrees.

math.AC↗
arXiv2025

Generalizing blocking semiovals in finite projective planes

Blocking semiovals and the determination of their (minimum) sizes constitute one of the central research topics in finite projective geometry. In this article we introduce the concept of blocking set with the $r_\infty$-property in a finite projective plane $\text{PG}(2,q)$, with $r_\infty$ a line of $\text{PG}(2,q)$ and $q$ a prime power. This notion greatly generalizes that of blocking semioval. We address the question of determining those integers $k$ for which there exists a blocking set of size $k$ with the $r_\infty$-property. To solve this problem, we build new theory which deeply analyzes the interplay between blocking sets in finite projective and affine planes.

math.CO↗
arXiv2025

Principal vector-spread Borel ideals

We study the class of squarefree principal vector-spread Borel ideals. We compute the minimal primary decomposition of these ideals and thereby we prove that they are sequentially Cohen-Macaulay. As the final conclusion of our results, we completely classify the ideals in our class having the property that their ordinary and symbolic powers coincide.

math.AC↗
arXiv2025

Comparison of stability indices of powers of graded ideals

In this paper, we compare the index of ass-stability $\text{astab}(I)$ and the index of $\text{v}$-stability $\text{vstab}(I)$ of powers of a graded ideal $I$. We prove that $\text{astab}(I)=1\le\text{vstab}(I)$ for any graded ideal $I$ in a 2-dimensional polynomial ring, and that $\text{vstab}(I)$ can be any positive integer in this situation. Moreover, given any integers $a,b\ge1$, we construct a graded ideal $I$ in a $3(a+1)$-dimensional polynomial ring such that $(\text{astab}(I),\text{vstab}(I))=(a,b)$.

math.AC↗
arXiv2025

Monomial ideals whose all matching powers are Cohen-Macaulay

In the present paper, we aim to classify monomial ideals whose all matching powers are Cohen-Macaulay. We especially focus our attention on edge ideals. The Cohen-Macaulayness of the last matching power of an edge ideal is characterized, providing an algebraic analogue of the famous Tutte theorem regarding graphs having a perfect matching. For chordal graphs, very well-covered graphs and Cameron-Walker graphs, we completely solve our problem.

math.AC↗
arXiv2025

The homological shift algebra of a monomial ideal

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. In this paper, we introduce the $i$th \textit{homological shift algebras} $\text{HS}_i(\mathcal{R}(I))=\bigoplus_{k\ge1}\text{HS}_i(I^k)$ of $I$. If $I$ has linear powers, these algebras have the structure of a finitely generated bigraded module over the Rees algebra $\mathcal{R}(I)$ of $I$. Hence, many invariants of $\text{HS}_i(I^k)$, such as depth, associated primes, regularity, and the $\text{v}$-number, exhibit well behaved asymptotic behavior. We determine several families of monomial ideals $I$ for which $\text{HS}_i(I^k)$ has linear resolution for all $k\gg0$. Finally, we show that $\text{HS}_i(I^k)$ is Golod for all monomial ideals $I\subset S$ with linear powers and all $k\gg0$.

math.AC↗
arXiv2025

Edge ideals and their asymptotic syzygies

Let $G$ be a finite simple graph, and let $I(G)$ denote its edge ideal. In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals $\text{HS}_i(I(G)^k)$. We introduce the notion of the $i$th homological strong persistence property for monomial ideals $I$, providing an algebraic characterization that ensures the chain of inclusions $\text{Ass}\,\text{HS}_i(I)\subseteq\text{Ass}\,\text{HS}_i(I^2)\subseteq\text{Ass}\,\text{HS}_i(I^3) \subseteq\cdots$. We prove that edge ideals possess both the $0$th and $1$st homological strong persistence properties. To this end, we explicitly describe the first homological shift algebra of $I(G)$ and show that $\text{HS}_1(I(G)^{k+1}) = I(G) \cdot \text{HS}_1(I(G)^k)$ for all $k \ge 1$. Finally, we conjecture that if $I(G)$ has a linear resolution, then $\text{HS}_i(I(G)^k)$ also has a linear resolution for all $k \gg 0$, and we present partial results supporting this conjecture.

math.AC↗
arXiv2025

Asymptotic behaviour of integer programming and the $\text{v}$-function of a graded filtration

The $\text{v}$-function of a graded filtration $\mathcal{I}=\{I_{[k]}\}_{k\ge0}$ is introduced. Under the assumption that $\mathcal{I}$ is Noetherian, we prove that the $\text{v}$-function $\text{v}(I_{[k]})$ is an eventually quasi-linear function. This result applies to several situations, including ordinary powers, and integral closures of ordinary powers, among others. As another application, we investigate the asymptotic behaviour of certain integer programming problems. Finally, we present the \textit{Macaulay2} package $\texttt{VNumber}$.

math.AC↗
arXiv2025

Algebraic study on permutation graphs

Let $G$ be a permutation graph. We show that $G$ is Cohen-Macaulay if and only if $G$ is unmixed and vertex decomposable. When this is the case, we obtain a combinatorial description for the $a$-invariant of $G$. Moreover, we characterize the Gorenstein permutation graphs.

math.AC↗