arXiv subjects
Publications and source records attributed to Aldo Conca.
An Artinian ideal $I$ of $k[x,y]$ has many Hilbert-Burch matrices. We show that there is a canonical choice. As an application, we determine the dimension of certain affine Gröbner cells and their Betti strata recovering results of Ellingsrud and Strømme, Göttsche and Iarrobino.
We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute the symmetric algebraic shifted complex of two disjoint simplicial complexes as was conjectured by Kalai. This result is the symmetric analogue of a theorem of Nevo who determined the exterior algebraic shifted complex of two disjoint simplicial complexes as predicted by Kalai.
This is a survey article on Gorenstein initial complexes of extensively studied ideals in commutative algebra and algebraic geometry. These include defining ideals of Segre and Veronese varieties, toric deformations of flag varieties known as Hibi ideals, determinantal ideals of generic matrices of indeterminates, and ideals generated by Pfaffians of generic skew symmetric matrices. We give a summary of recent work on the construction of squarefree Gorenstein initial ideals of these ideals when the ideals are themselves Gorenstein. We also present our own independent results for the Segre, Veronese, and some determinantal cases.
Let $K$ be an infinite field and $R=K[x_1,...,x_n]$ be the polynomial ring. Let $V=V_1, ..., V_m$ be a collection of vector spaces of linear forms. Denote by $A(V)$ the $K$-subalgebra of $R$ generated by the elements of the product $V_1... V_m$. Our goal is to investigate the properties of the algebra $A(V)$ and the relations with two problems in algebraic combinatorics White's and related conjectures on polymatroids and the study of integral posets.
Let I be the defining ideal of a smooth irreducible complete intersection space curve C with defining equations of degrees a and b. We use the partial elimination ideals introduced by Mark Green to show that the lexicographic generic initial ideal of I has Castelnuovo-Mumford regularity 1+ab(a-1)(b-1)/2 with the exception of the case a=b=2, where the regularity is 4. Note that ab(a-1)(b-1)/2 is exactly the number of singular points of a general projection of C to the plane. Additionally, we show that for any term ordering tau, the generic initial ideal of a generic set of points in P^r is a tau-segment ideal.
By definition, an $\m$-primary ideal $I$ in a 2-dimensional regular local ring $(R, \m)$ is contracted if $I=R \cap IR[\m/x]$ for some $x \in \m \setminus \m^2$. Contracted ideals have been introduced by Zariski and used for proving the unique factorization theorem for complete (i.e. integrally closed) ideals. Any complete ideal is contracted but not the other way round. While the associated graded rings to complete ideals are always Cohen-Macaulay, this is not the case for contracted ideals. Our goal is to study depth, Hilbert function and defining equations of the graded rings of homogeneous contracted ideals. We show, by using quadratic transform, that the depth of the associated graded ring to a contracted ideal $I$ is determined by depth of the associated graded rings to a certain family of monomial ideals (indeed lex-segments) which are naturally attached to $I$. For certain classes of contracted ideals we show that the associated graded ring is Cohen-Macaulay or at least has positive depth.
Let $X$ be an $n\times m$ matrix of indeterminates over a field $K$ (of sufficiently large characteristic) and $M_t$ the set of $m$-minors of $X$. We consider two objects: (1) the Ress algebra of the polynomial ring $K[X]$ with respect to the ideal $I_t$ generated by $M_t$, and (2) the $A_t$ subalgebra of $K[X]$ generated by $M_t$. Note that $A_t$ is tHE coordinate ring of a Grassmannian if $t=\min(m,n)$; also the cases $t=1$ and $t=m-1=n-1$ are easily understood, since $A_t$ is a polynomial ring over $K$ in these cases. For both objects we compute the divisor class group and the canonical class. In particular we determine the Gorenstein rings among the $A_t$. It turns out that $A_t$ is Gorenstein exactly in the cases listed above and when $t(m+n)=mn$. We use initial methods, based on the straightening law and KRS. They can be applied to other types of determinantal ideals, too. We do this explicitly for generic Hankel matrices.
The Castelnuovo-Mumford regularity $\reg(I)$ is one of the most important invariants of a homogeneous ideal $I$ in a polynomial ring. A basic question is how the regularity behaves with respect to taking powers of ideals. It is known that in the long-run $\reg(I^k)$ is a linear function of $k$. We show that in the short-run the regularity of $I^k$ can be quite "irregular". For any given integer $d>1$ we construct an ideal $J$ generated by $d+5$ monomials of degree $d+1$ in 4 variables such that $\reg(J^k)=k(d+1)$ for every $k<d$ and $\reg(J^d)\geq d(d+1)+d-1$.
We give an introduction to the theory of initial ideals and initial algebras with emphasis on the transfer of structural properties.
In the first part of the paper we answer (positively) a question raised by the first author which has to do with some sort of rigity of the tail of resolution of an ideal. Let $I$ be a homogeneous ideal in a polynomial ring over a field of characteristic 0. Denote by $β_i(I)$ the $i$-th Betti number of $I$ and by $Gin(I)$ the revlex generic initial ideal of $I$. In general one has $β_i(I)\leq β_i(Gin(I))$ and we show that if $β_i(I)=β_i(Gin(I))$ for some $i$ then $β_j(I)=β_j(Gin(I))$ for all $j>i$. In the second part of the paper we answer a question of Eisenbud and Huneke. We prove that if $I$ is $m$-primary and $I\subset m^d$ then $β_i(m^d)\leq β_i(Gin(I))$ for all $i$.
In a polynomial ring $R$ with $n$ variables, for every homogeneous ideal $I$ and for every $p\leq n$ we consider the Koszul homology $H_i(p,R/I)$ with respect to a sequence of $p$ of generic linear forms and define the Koszul-Betti number $β_{ijp}(R/I)$ of $R/I$ to be the dimension of the degree $j$ part of $H_i(p,R/I)$. In characteristic 0, we show that the Koszul-Betti numbers of any ideal $I$ are bounded above by those of any gin of $I$ and also by those of the Lex-segment of $I$. We also investigate the set $Gins(I)$ of all the gin of $I$ and show that the Koszul-Betti numbers of any ideal in $Gins(I)$ are bounded below by those of the gin-revlex of $I$ and present examples showing that in general there is no $J$ is $Gins(I)$ such that the Koszul-Betti numbers of any ideal in $Gins(I)$ are bounded above by those of $J$.
The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that Jm^k=m^{k+1}. Vasconcelos conjectured that the reduction number of A=R/I can only increase by passing to the initial ideal, i.e r(R/I)\leq r(R/in(I)). The goal of this note is to prove the conjecture.
We discuss the behavior of the Castelnuovo-Mumford regularity under certain operations on ideals and modules, like products or powers. In particular, we show that reg(IM) can be larger than reg(M)+reg(I) even when I is an ideal of linear forms and M is a module with a linear resolution. On the other hand, we show that any product of ideals of linear forms has a linear resolution. We also discuss the case of polymatroidal ideals and show that any product of determinantal ideals of a generic Hankel matrix has a linear resolution.
The first sections contain a survey of the application of the Knuth-Robinson-Schensted corerspondence to the computation of Groebner bases of determinantal ideals. We also set up a conceptual framework for this application in terms of so-called "KRS invariants". Then we show that the initial ideal of a determinantal ideal "defined by shape" is given by its KRS image. We furthermore characterize those among these ideals that even have a Groebner basis of products of minors, and show that they can be characterized in terms of Greene's KRS invariants. Furthermore it is shown that for the ideal generated by all t-minors the formation of initial ideal and symbolic power commutes. The last section contains a discussion of potential KRS invariants related to so-called 1-cogenerated ideals.