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Adela Vraciu

Publications and source records attributed to Adela Vraciu.

At least 19 recordsLinked to original sources
arXiv2026

The generators of a colon ideal with an application to the weak Lefschetz property for monomial almost complete intersections in three variables

Much progress has been made in classifying when the weak Lefschetz property holds for $A=\mathbb{F}[x,y,z]/I$ where $\text{char}(\mathbb{F})=0$ and $I=(x^{d_{1}},y^{d_{2}},z^{d_{3}},x^{a_{1}}y^{a_{2}}z^{a_{3}})$ is a monomial almost complete intersection. We connect this problem to the setting of two variables through a certain relation. In so doing, we are led to determine explicit formulas for the generators of the colon ideal $(x^{d_{1}},y^{d_{2}}):(x+y)^{a_{3}}$. With these generators in hand, we construct a matrix and show that failure of WLP for $A$ is dictated by the vanishing of a certain polynomial (namely the determinant of our matrix) when $A$ is level. We further show in the level case that a conjecture first posed by Migliore, Miró-Roig, and Nagel is true in a few new cases.

math.AC↗
arXiv2026

A study of maximal shifts in the minimal graded free resolution of the residue field

We study two invariants, rate and slant, that describe the extremal behavior of the maximal shifts in the minimal free resolution of the residue field over a standard graded algebra. We provide bounds and computations of these invariants for a variety of rings, including compressed level algebras, rings defined by general forms and rings defined by monomials. We also prove an asymptotic additivity property of the maximal shifts for certain classes of rings, including Golod rings, which allows to interpret slant as a limit.

math.AC↗
arXiv2025

On the weak Lefschetz property for ideals generated by powers of general linear forms

We provide a description of initial ideals for almost complete intersections generated by powers of general linear forms and prove that WLP in a fixed degree $d$ holds when the number of variables $n$ is sufficiently large compared to $d$. In particular, we show that if $n\geq 3d-2$ then WLP holds for the ideal generated by squares at the degree $d$ spot and for $n\ge \frac{3d-3}{2}$ WLP holds for ideal generated by cubes at the degree $d$ spot. Finally, we prove that WLP fails for the ideal generated by squares when $n< 3d -2$ at the $d$th spot by finding an explicit element in the kernel of the multiplication by a general linear form. This shows that our bound on $n$ is sharp in the case of the squares.

math.AC↗
arXiv2024

Rings for which general linear forms are exact zero divisors

We investigate the standard graded $k$-algebras over a field $k$ of characteristic zero for which general linear forms are exact zero divisors. We formulate a conjecture regarding the Hilbert function of such rings. We prove our conjecture in the case when the ring is a quotient of a polynomial ring by a monomial idea, and also in the case when the ideal is generated in degree 2 and all but one of the generators are monomials.

math.AC↗
arXiv2022

Values of the F-pure threshold for homogeneous polynomials

We find a formula, in terms of n, d and p, for the value of the F-pure threshold for the generic homogeneous polynomial of degree d in n variables over an algebraically closed field of characteristic p. We also show that, in every characteristic p and for all d (greater than 3) not divisible by p, there always exist reduced polynomials of degree d whose F-pure threshold is a truncation of the base p expansion of 2/d at some place; in particular, there always exist reduced polynomials whose F-pure threshold is strictly less than 2/d. We provide an example to resolve, negatively, a question proposed by Hernandez, Núñez-Betancourt, Witt and Zhang, as to whether a list of necessary restrictions they prove on the F-pure threshold of reduced forms are "minimal" for large p. On the other hand, we also provide evidence supporting and refining their ideas, including identifying specific truncations of the base p expansion of 2/d that are always F-pure thresholds for reduced forms of degree d, and computations that show their conditions suffice (in every characteristic) for degrees up to eight and several other situations. Finally, we point out a lower bound on the F-pure threshold of a reduced form in terms of its degree and the characteristic p.

math.AC↗
arXiv2022

Lower Bounds on the F-pure Threshold and Extremal Singularities

We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$ if and only if $f\in \mathfrak m^{[q]}$ and $d=q+1$, where $q$ is a power of $p$. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such "extremal singularities," and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are "extremal," for example, in terms of the configurations of lines they can contain.

math.AC↗
arXiv2022

Classification of Frobenius Forms in five variables

We classify Frobenius forms, a special class of homogeneous polynomials in characteristic $p>0$, in up to five variables over an algebraically closed field. We also point out some of the similarities with quadratic forms.

math.AC↗
arXiv2022

Cubic Surfaces of Characteristic Two

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a "triangle".

math.AC↗
arXiv2019

On the existence of non-free totally reflexive modules

For a standard graded Cohen-Macaulay ring $S$, if the quotient $S/(\underline{x})$ admits non-free totally reflexive modules, where $\underline{x}$ is a system of parameters consisting of elements of degree one, then so does the ring $S$. As an application, we consider the question of which Stanley-Reisner rings of graphs admit non-free totally reflexive modules.

math.AC↗
arXiv2017

Poincaré series of compressed local Artinian rings with odd top socle degree

We define a notion of compressed local Artinian ring that does not require the ring to contain a field. Let $(R,\mathfrak m)$ be a compressed local Artinian ring with odd top socle degree $s$, at least five, and $\operatorname{socle}(R)\cap \mathfrak m^{s-1}=\mathfrak m^s$. We prove that the Poincaré series of all finitely generated modules over $R$ are rational, sharing a common denominator, and that there is a Golod homomorphism from a complete intersection onto $R$.

math.AC↗
arXiv2017

Totally reflexive modules over rings that are close to Gorenstein

Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the canonical module of $R$ has a direct summand $T$ which is a test module for freeness over $R$ in the sense that if $\mathrm{Tor}_+^R(T,N)=0$, for some finitely generated $R$-module $N$, then $N$ is free.

math.AC↗
arXiv2015

On the degrees of relations on $x_1^{d_1}, \ldots, x_n^{d_n}, (x_1+ \ldots + x_n)^{d_{n+1}}$ in positive characteristic

We give a formula for the smallest degree of a non-Koszul relation on $x_1^{d_1}, \ldots, x_n^{d_n}, (x_1+\ldots +x_n)^{d_{n+1}}\in k[x_1, \ldots, x_n]$ (under certain assumptions on $d_1, \ldots, d_{n+1}$) where $k$ is a field of positive characteristic $p$. As an application of our result, we give a formula for the diagonal F-threshold of a diagonal hypersurface. Another application is a characterization, depending on the characteristic $p$ of $k$, of the values of $d_1, \ldots, d_{n+1}$ (satisfying certain assumptions) such that the ring $k[x_1, \ldots, x_{n+1}]/(x_1^{d_1}, \ldots, x_{n+1}^{d_{n+1}})$ has the weak Lefschetz property.

math.AC↗
arXiv2015

Minimal quasi-complete intersection ideals

A quasi-complete intersection (q.c.i.) ideal of a local ring is an ideal with "free exterior Koszul homology"; the definition can also be understood in terms of vanishing of André-Quillen homology functors. Principal q.c.i. ideals are well understood, but few constructions are known to produce q.c.i. ideals of grade zero that are not principal. This paper examines the structure of q.c.i. ideals. We exhibit conditions on a ring $R$ which guarantee that every q.c.i. ideal of $R$ is principal. On the other hand, we give an example of a minimal q.c.i. deal $I$ which does not contain any principal q.c.i. ideals and is not embedded, in the sense that no faithfully flat extension of $I$ can be written as a quotient of complete intersection ideals. We also describe a generic situation in which the maximal ideal of $R$ is an embedded q.c.i. ideal that does not contain any principal q.c.i. ideals.

math.AC↗
arXiv2014

Special tight closure

We prove that in normal rings the tight closure of an ideal can be computed as the sum of the ideal and a piece of the tight closure, called the special tight closure.

math.AC↗
arXiv2013

Exact pairs of homogeneous zero divisors

Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that if S is a compressed level algebra, then S does not contain any homogeneous zero divisors.

math.AC↗