Related papers: The Lefschetz properties of monomial complete inte…
We study the WLP and SLP of artinian monomial ideals in $S=\mathbb{K}[x_1,\dots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at…
In this article, we consider the weighted generating function of matchings in the complete graph. We define an Artinian Gorenstein algebra as the quotient ring of a polynomial ring by the annihilator of the generating function. We show the…
It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded…
Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,\ldots, x_n]$. We show that if either: 1) $I$ is almost complete intersection, 2) $I$ can be generated by less than four monomials; or 3) $I$ is the Stanley-Reisner…
In 2012, Migliore, the first author, and Nagel conjectured that, for all $n\geq 4$, the artinian ideal $I=(L_0^d,\ldots,L_{2n+1}^d) \subset R=k[x_0,\ldots,x_{2n}]$ generated by the $d$-th powers of $2n+2$ general linear forms fails to have…
We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this…
We give a Belyi-type characterisation of smooth complete intersections of general type over $\mathbb{C}$ which can be defined over $\bar{\mathbb{Q}}$. Our proof uses the higher-dimensional analogue of the Shafarevich boundedness conjecture…
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras -…
In this paper, we investigate the weak Lefschetz property for tensor products of Artinian monomial algebras and complete quadratic monomial algebras. As an application, we classify the weak Lefschetz property of the Artinian algebras…
We give a sharp lower bound for the Hilbert function in degree $d$ of artinian quotients $\Bbbk[x_1,\ldots,x_n]/I$ failing the Strong Lefschetz property, where $I$ is a monomial ideal generated in degree $d \geq 2$. We also provide sharp…
We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is…
We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally…
It is well known that for a subscheme $V$ in ${\mathbb P}^{n}$ of codimension two, the conditions (1) $V$ is ACM, and (2) $V$ is "licci" (i.e. $V$ is in the liaison class of a complete intersection) are equivalent. In higher codimension,…
In his preprint ``Differential-Geometric Characterizations of Complete Intersections'' (alg-geom/9407002), J.M.Landsberg introduces an elementary characterization of complete intersections. The proof of this criterion uses the method of…
The theory of Rees algebras of monomial ideals has been extensively studied, and as a consequence, many (sometimes partial) equivalences between algebraic properties of monomial ideals, and combinatorial properties of simplicial complexes…
Let $A$ be a graded complete intersection over a field and $B$ the monomial complete intersection with the generators of the same degrees as $A$. The EGH conjecture says that if $I$ is a graded ideal in $A$, then there should be an ideal…
We present an alternate proof of a result of F\'eray and Reiner characterizing posets whose $P$-partition rings are complete intersections. This shortened proof relates the complete intersection property to a simple structural property of a…
The "finite intersection property" for bifunctions is introduced and its relationship with generalized monotonicity properties is studied. Some results concerning existence of solution for (quasi-)equilibrium problems are established and…
We classify the minimal number of generators of artinian equigenerated monomial ideals $I$ such that $\Bbbk[x_1,\ldots,x_n]/I$ is forced to have the weak Lefschetz property.
A combinatorial property of prositive group presentations, called completeness, is introduced, with an effective criterion for recognizing complete presentations, and an iterative method for completing an incomplete presentation. We show…