Mathematics

The notion of ε -multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that the relative ε -multiplicity of reduced Noetherian graded algebras over an excellent local ring exists as a limit. We also obtain a generalization of Cutkosky's result concerning ε -multiplicity, as a corollary of our main theorem.

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove that negative curves are rational in many cases. We also see that the Cox ring of the blow-up of a toric variety at the point (1,1,...,1) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14th problem) coincides with the Cox ring of some normal projective variety.

In arXiv:math/0405373 , Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of the embedding of a projective space P n−1 ↪ P r−1 determined by generators of a linearly presented m -primary ideal. This result implies in particular that the image is scheme defined by equations of degree at most n . In this text we prove that the ideal of maximal minors of the Jacobian dual matrix associated to the input ideal defines the image as a scheme; it is generated in degree n . Showing that this ideal has a linear resolution would imply that the conjecture in arXiv:math/0405373 holds. Furthermore, if this ideal of minors coincides with the one of the image in degree n - what we hope to be true - the linearity of the resolution of this ideal of maximal minors is equivalent to the conjecture in arXiv:math/0405373.

We focus on the structure of a homogeneous Gorenstein ideal I of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that I is generated in a fixed degree d . For such an ideal I this degree comes along with the minimal number of generators of I and the degree of the entries of the associated skew-symmetric matrix in a simple formula. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal I of codimension three filling them. We conjecture that, for arbitrary n≥2 , an ideal $I\subset \kk[x_1,\ldots,x_n]$ generated by a general set of r≥n+2 forms of degree d≥2 is Gorenstein if and only if d=2 and r=( n+1 2 )−1 . We prove the `only if' implication of this conjecture when n=3 . For arbitrary n≥2 , we prove that if d=2 and r≥(n+2)(n+1)/6 then the ideal is Gorenstein if and only if r=( n+1 2 )−1 , which settles the `if' assertion of the conjecture for n≤5 . Finally, we give a partial answer to one of the questions of Fröberg--Lundqvist. In a different direction, we reveal a connection between the Macaulay inverse and the so-called Newton dual, a matter so far not brought out to our knowledge. Finally, we consider the question as to when the link ( ℓ m 1 ,…, ℓ m n ):f is equigenerated, where ℓ 1 ,…, ℓ n are independent linear forms and f is a form, is given a solution in some important cases.

The overall goal is to approach the Cohen--Macaulay property of the special fiber F(I) of an equigenerated homogeneous ideal I in a standard graded ring over an infinite field. When the ground ring is assumed to be local, the subject has been extensively looked at. Here, with a focus on the graded situation, one introduces two technical conditions, called respectively, {\em analytical tightness} and {\em analytical adjustment}, in order to approach the Cohen--Macaulayness of F(I) . A degree of success is obtained in the case where I in addition has analytic deviation one, a situation looked at by several authors, being essentially the only interesting one in dimension three. Naturally, the paper has some applications in this case.

Let R be a commutative ring with identity and let I be a two-generated ideal of R . We denote by SL 2 (R) the group of 2×2 matrices over R with determinant 1 . We study the action of SL 2 (R) by matrix right-multiplication on V 2 (I) , the set of generating pairs of I . Let Fitt 1 (I) be the second Fitting ideal of I . Our main result asserts that V 2 (I)/ SL 2 (R) identifies with a group of units of R/ Fitt 1 (I) via a natural generalization of the determinant if I can be generated by two regular elements. This result is illustrated in several Bass rings for which we also show that SL n (R) acts transitively on V n (I) for every n>2 . As an application, we derive a formula for the number of cusps of a modular group over a quadratic order.

Let R be a commutative ring with identity and let M be an R -module which is generated by μ elements but not fewer. We denote by SL n (R) the group of the n×n matrices over R with determinant 1 . We denote by E n (R) the subgroup of SL n (R) generated by the the matrices which differ from the identity by a single off-diagonal coefficient. Given n≥μ and G∈{ SL n (R), E n (R)} , we study the action of G by matrix right-multiplication on V n (M) , the set of elements of M n whose components generate M . Assuming that M is finitely presented and that R is an elementary divisor ring or an almost local-global coherent Prüfer ring, we obtain a description of V n (M)/G which extends the author's earlier result on finitely generated modules over quasi-Euclidean rings.

Given a generically finite local extension of valuation rings V?�W , the question of whether W is the localization of a finitely generated V -algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when W is essentially of finite type over V in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic 0 using embedded resolution of singularities. In this paper we prove Knaf's conjecture in full generality.

The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of Nakayama's lemma; complete theories of minimal and dense primary, secondary, and irreducible decomposition, including associated and attached faces; socles and tops; minimality and density for downset hulls, upset covers, and fringe presentations; Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic or piecewise-linear (PL) have the relevant property preserved by functorial constructions as well as by minimal primary and secondary decompositions. And when the modules in question are subquotients of the group itself, such as monomial ideals and quotients modulo them, minimal primary and secondary decompositions are canonical, as are irreducible decompositions up to the new real-exponent notion of density.

We investigate the problem of representation of moment sequences by measures in Polynomial Optimization Problems, consisting in finding the infimum f * of a real polynomial f on a real semialgebraic set S defined by a quadratic module Q. We analyse the exactness of Moment Matrix (MoM) relaxations, dual to the Sum of Squares (SoS) relaxations, which are hierarchies of convex cones introduced by Lasserre to approximate measures and positive polynomials. We show that the MoM relaxation coincides with the dual of the SoS relaxation extended with the real radical of the support of the associated quadratic module Q. We prove that the vanishing ideal of the semialgebraic set S is generated by the kernel of the Hankel operator associated to a generic element of the truncated moment cone for a sufficiently high order of the MoM relaxation. When the quadratic module Q is Archimedean, we show the convergence, in Hausdorff distance, of the convex sets of the MoM relaxations to the convex set of probability measures supported on S truncated in a given degree. We prove the exactness of MoM relaxation when S is finite and when regularity conditions, known as Boundary Hessian Conditions, hold on the minimizers. This implies that MoM exactness holds generically. When the set of minimizers is finite, we describe a MoM relaxation which involves f * , show its MoM exactness and propose a practical algorithm to achieve MoM exactness. We prove that if the real variety of polar points is finite then the MoM relaxation extended with the polar constraints is exact. Effective numerical computations illustrate this MoM exactness property.