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Aldo Conca

Publications and source records attributed to Aldo Conca.

At least 55 records · Page 3Linked to original sources
arXiv2013

Koszul algebras and their syzygies

These are the notes of the lectures of the author at the 2013 CIME/CIRM summer school on Combinatorial Algebraic Geometry. Koszul algebras, introduced by Priddy, are positively graded K-algebras R whose residue field K has a linear free resolution as an R-module. The first part of the notes is devoted to the introduction of Koszul algebras and their characterization in terms of Castelnuovo-Mumford regularity. In the second part we discuss recernt results on the syzygies of Koszul algebras. Finally in the last part we discuss the Koszul property of Veronese algebras and of algebras associated with collections of hyperspaces.

math.AC↗
arXiv2013

Subadditivity of syzygies of Koszul algebras

Estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings. For a class that includes Koszul algebra in almost all characteristics, these degrees are shown to increase by at most 2 from one syzygy module to the next one. Even slower growth is proved if, in addition, the algebra satisfies Green and Lazarsfeld's condition N_q with q > 1.

math.AC↗
arXiv2013

Relations between the minors of a generic matrix

It is well-known that the Plücker relations generate the ideal of relations of the maximal minors of a generic matrix. In this paper we discuss the relations between minors of a (non-maximal) fixed size. We will exhibit minimal relations in degrees 2 (non-Plücker in general) and 3, and give some evidence for our conjecture that we have found the generating system of the ideal of relations. The approach is through the representation theory of the general linear group.

math.AC↗
arXiv2013

Ideals generated by superstandard tableaux

We investigate products J of ideals of "row initial" minors in the polynomial ring K[X] defined by a generic m-by-n matrix. Such ideals are shown to be generated by a certain set of standard bitableaux that we call superstandard. These bitableaux form a Gröbner basis of J, and J has a linear minimal free resolution. These results are used to derive a new generating set for the Grothendieck group of finitely generated (T_m x GL_n(K))-equivariant modules over K[X]. We employ the Knuth--Robinson--Schensted correspondence and a toric deformation of the multi-Rees algebra that parameterizes the ideals J.

math.AC↗
arXiv2013

Universal Groebner bases for maximal minors

A set of polynomials G in a polynomial ring S over a field is said to be a universal Groebner basis, if G is a Groebner basis with respect to every term order on S. Twenty years ago Bernstein, Sturmfels, and Zelevinsky proved that the set of the maximal minors of a matrix X of variables is a universal Groebner basis. Boocher recently proved that any initial ideal of the ideal of maximal minors of X has a linear resolution. In this paper we give a quick proof of the results mentioned above. Our proof is based on a specialization argument. Then we show that similar statements hold in a more general setting, for matrices of linear forms satisfying certain homogeneity conditions. More precisely, we show that the set of maximal minors of a matrix L of linear forms is a universal Groebner basis for the ideal I that it generates, provided that L is column-graded. Under the same assumption we show that every initial ideal of I has a linear resolution. Furthermore, the projective dimension of I and of its initial ideals is n-m, unless I=0 or a column of L is identically 0. Here L is a matrix of size m times n, and m is smaller than or equal to n. If instead L is row-graded, then we prove that I has a universal Groebner basis of elements of degree m and that every initial ideal of I has a linear resolution, provided that I has the expected codimension. The proofs are based on a rigidity property of radical Borel fixed ideals in a multigraded setting: We prove that if two Borel fixed ideals I and J have the same multigraded Hilbert series and I is radical, then I = J. We also discuss some of the consequences of this rigidity property.

math.AC↗
arXiv2013

Maximal minors and linear powers

An ideal I in a polynomial ring S has linear powers if all the powers I^k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers.

math.AC↗
arXiv2012

Koszul property of projections of the Veronese cubic surface

Let V be the Veronese cubic surface in P^9. We classify the projections of V to P^8 whose coordinate rings are Koszul. In particular we obtain a purely theoretical proof of the Koszulness of the pinched Veronese, a result obtained originally by Caviglia using filtrations, deformations and computer assisted computations. To this purpose we extend, to certain complete intersections, results of Conca, Herzog, Trung and Valla concerning homological properties of diagonal algebras.

math.AC↗
arXiv2012

Koszul algebras and regularity

This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford regularity. We describe several techniques to establish the Koszulness of algebras. We discuss variants of the Koszul property such as strongly Koszul, absolutely Koszul and universally Koszul. We present several open problems related with these notions and their local variants.

math.AC↗
arXiv2012

New free divisors from old

We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free divisors, or divisors with triangular discriminant matrix. We show how to create families of quasihomogeneous free divisors through the chain rule or by extending them into the tangent bundle. We also discuss whether general divisors can be extended to free ones by adding components and show that adding a normal crossing divisor to a smooth one will not succeed.

math.AC↗
arXiv2012

Regularity bounds for Koszul cycles

We study the module of Koszul cycles $Z_t(I,M)$ of a homogeneous ideal $I$ in a polynomial ring $S$ with respect to a graded module $M$. Under mild assumptions on the base field we prove that the regularity of $Z_t(I,S)$ is a subadditive function of the homological position t when I is 0-dimensional. For Borel-fixed ideals $I$ and $J$ we prove that the regularity of $Z_t(I,S/J)$ is bounded above by $t(1+\reg I)+\reg S/J$.

math.AC↗
arXiv2010

Koszul cycles

We prove regularity bounds for Koszul cycles holding for every ideal of dimension at most 1 in a polynomial ring. We generalize the lower bound for the Green-Lazarsfeld index of Veronese rings we proved in arXiv:0902.2431 to the multihomogeneous setting.

math.AC↗
arXiv2009

Koszul homology and syzygies of Veronese subalgebras

A graded K-algebra R has property N_p if it is generated in degree 1, has relations in degree 2 and the syzygies of order less or equal to p on the relations are linear. The Green-Lazarsfeld index of R is the largest p such that it satisfies the property N_p. Our main results assert that (under a mild assumption on the base field K) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index greater or equal to c+1. The same conclusion also holds for an arbitrary standard graded algebra, provided c is sufficiently large.

math.AC↗
arXiv2009

Free resolutions over commutative Koszul algebras

For R=Q/J with Q a commutative graded algebra over a field and J non-zero, we relate the slopes of the minimal resolutions of R over Q and of k=R/R_{+} over R. When Q and R are Koszul and J_1=0 we prove Tor^Q_i(R,k)_j=0 for j>2i, for each non-negative integer i, and also for j=2i when i>dim Q-dim R and pd_QR is finite.

math.AC↗
arXiv2009

Integrally closed and componentwise linear ideals

In two dimensional regular local rings integrally closed ideals have a unique factorization property and have a Cohen-Macaulay associated graded ring. In higher dimension these properties do not hold for general integrally closed ideals and the goal of the paper is to identify a subclass of integrally closed ideals for which they do. We restrict our attention to 0-dimensional homogeneous ideals in polynomial rings $R$ of arbitrary dimension and identify a class of integrally closed ideals, the Goto-class $\G^*$, that is closed under product and that has a suitable unique factorization property. Ideals in $\G^*$ have a Cohen-Macaulay associated graded ring if either they are monomial or $\dim R\leq 3$. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals.

math.AC↗
arXiv2008

The variety of exterior powers of linear maps

Let $K$ be a field and $V$ and $W$ be $K$-vector spaces of dimension $m$ and $n$. Let $ϕ$ be the canonical map from $Hom(V,W)$ to $Hom(\wedge^t V,\wedge^t W)$. We investigate the Zariski closure $X_t$ of the image $Y_t$ of $ϕ$. In the case $t=\min(m,n)$, $Y_t=X_t$ is the cone over a Grassmannian, but $X_t$ is larger than $Y_t$ for $1<t<\min(m,n)$. We analyze the $G=\GL(V)\times\GL(W)$-orbits in $X_t$ via the corresponding $G$-stable prime ideals. It turns out that they are classified by two numerical invariants, one of which is the rank and the other a related invariant that we call small rank. Surprisingly, the orbits in $X_t\setminus Y_t$ arise from the images $Y_u$ for $u<t$ and simple algebraic operations. In the last section we determine the singular locus of $X_t$. Apart from well-understood exceptional cases, it is formed by the elements of rank $\le 1$ in $Y_t$.

math.AC↗
arXiv2008

Groebner bases for spaces of quadrics of codimension 3

Let $R=\oplus_{i\geq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $\dim R_2\leq 3$ and that $K$ is algebraically closed of characteristic $\neq 2$. We show that $R$ is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.

math.AC↗
arXiv2007

Contracted ideals and the Groebner fan of the rational normal curve

The paper has two goals: the study the associated graded ring of contracted homogeneous ideals in $K[x,y]$ and the study of the Groebner fan of the ideal $P$ of the rational normal curve in ${\bf P}^d$. These two problems are, quite surprisingly, very tightly related. We completely classify the contracted ideals with a Cohen-Macaulay associated graded rings in terms of the numerical invariants arising from Zariski's factorization. We determine explicitly all the initial ideals (monomial or not) of $P$ that are Cohen-Macaulay.

math.AC↗