Independence numbers of graphs and generators of ideals

This article investigates the generators of certain homogeneous ideals which are associated with graphs with bounded independence numbers. These ideals first appeared in the theory oft-designs. The main theorem suggests a new approach to the Clique Problem which isNP-complete. This theorem has a more general form in commutative algebra dealing with ideals associated with unions of linear varieties. This general theorem is stated in the article; a corollary to it generalizes Turán’s theorem on the maximum graphs with a prescribed clique number. Research supported in part by NSF Grant MCS77-03533.     

Local uniformization and arc spaces

Let X be a variety over a perfect field k and let $X_{\infty }$ be its space of arcs. Given a closed subset Z of X, let $X_{\infty }^Z$ denote the subscheme of $X_{\infty }$ consisting of all arcs centered at some point of Z. We prove that Local Uniformization implies that $X_{\infty }^Z$ has a finite number of irreducible components for each closed subset Z of X. In particular, Local Uniformization implies that $X_{\infty }^{\mathit{Sing}X}$ has a finite number of irreducible components.     

Strongly irreducible ideals of a commutative ring

An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal.     

Stirling numbers and integer partitions

In this paper, we prove that the Stirling numbers of both kinds can be written as sums over integer partitions. As corollaries, we rewrite some identities with Stirling numbers of both kinds without Stirling numbers.     

A characteristic <Emphasis Type="Italic">p</Emphasis> analogue of plt singularities and adjoint ideals

We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair (X, Δ) is plt if and only if the modulo p reduction of (X, Δ) is divisorially F-regular for all large p ≫ 0. Here, divisorially F-regular pairs are a class of singularities in positive characteristic introduced by Hara and Watanabe (J Algebra Geom 11:363–392, 2002) in terms of Frobenius splitting. The author was partially supported by Grant-in-Aid for Young Scientists (B) 17740021 from JSPS.     

Properties of chains of prime ideals in an amalgamated algebra along an ideal

Let f:A→B be a ring homomorphism and let J be an ideal of B. In this paper, we study the amalgamation of A with B along J with respect to f (denoted by A?^fJ), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D’Anna and Fontana in 2007, and other classical constructions (such as the A+XB[X], the A+XB?X? and the D+M constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension.     

Generic tropical initial ideals of Cohen-Macaulay algebras

We study the generic tropical initial ideals of a positively graded Cohen-Macaulay algebra R over an algebraically closed field k. Building on work of Römer and Schmitz, we give a formula for each initial ideal, and we express the associated quasivaluations in terms of certain I-adic filtrations. As a corollary, we show that in the case that R is a domain, every initial ideal coming from the codimension 1 skeleton of the tropical variety is prime, so “generic presentations of Cohen-Macaulay domains are well-poised in codimension 1.”     

Uniformly primary ideals

This article introduces and advances the basic theory of “uniformly primary ideals” for commutative rings, a concept that imposes a certain boundedness condition on the usual notion of “primary ideal”. Characterizations of uniformly primary ideals are provided along with examples that give the theory independent value. Applications are also provided in contexts that are relevant to Noetherian rings.     

Globally F-regular and log Fano varieties

We prove that every globally F-regular variety is log Fano. In other words, if a prime characteristic variety X is globally F-regular, then it admits an effective Q-divisor Δ such that −KX−Δ is ample and (X,Δ) has controlled (Kawamata log terminal, in fact globally F-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non-Q-Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally F-regular type. Our techniques apply also to F-split varieties, which we show to satisfy a “log Calabi-Yau” condition. We also prove a Kawamata-Viehweg vanishing theorem for globally F-regular pairs.     

Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture

The hypersurfaces of degree d in the projective space Pn correspond to points of PN, where . Now assume d=2e is even, and let X(n,d)⊆PN denote the subvariety of two e-fold hyperplanes. We exhibit an upper bound on the Castelnuovo regularity of the ideal of X(n,d), and show that this variety is r-normal for r?2. The latter result is representation-theoretic, and says that a certain GLn+1-equivariant morphism

Sr(S2e(Cn+1))→S₂(Sre(Cn+1))     

Quasi-socle ideals and Goto numbers of parameters

Goto numbers for certain parameter ideals Q in a Noetherian local ring (A,m) with Gorenstein associated graded ring are explored. As an application, the structure of quasi-socle ideals I=Q:m^q (q≥1) in a one-dimensional local complete intersection and the question of when the graded rings are Cohen-Macaulay are studied in the case where the ideals I are integral over Q.     

Generalized cover ideals and the persistence property

Let I   be a square-free monomial ideal in R=k[_x1,…,_xn]$R=k[x_1,\dots ,x_n]$, and consider the sets of associated primes Ass(I^s)$\mathrm{Ass}(I^s)$ for all integers s?1$s?1$. Although it is known that the sets of associated primes of powers of I eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph G that generalizes the cover ideal construction. When G   is a tree, we explicitly determine Ass(I^s)$\mathrm{Ass}(I^s)$ for all s?1$s?1$. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.     

Chains of prime ideals in tensor products of algebras

This paper investigates the length of particular chains of prime ideals in tensor products of algebras over a field k. As an application, we compute dim(A⊗_kA) for a new family of domains A that are k-algebras.     

Nilpotent Lie algebras with 2-dimensional commutator ideals

We classify all (finitely dimensional) nilpotent Lie k-algebras h with 2-dimensional commutator ideals h^′, extending a known result to the case where h^′ is non-central and k is an arbitrary field. It turns out that, while the structure of h depends on the field k if h^′ is central, it is independent of k if h^′ is non-central and is uniquely determined by the dimension of h. In the case where k is algebraically or real closed, we also list all nilpotent Lie k-algebras h with 2-dimensional central commutator ideals h^′ and dim_kh?11.     

Factoring ideals in Prüfer domains

We show that in certain Prüfer domains, each nonzero ideal I can be factored as , where I^v is the divisorial closure of I and is a product of maximal ideals. This is always possible when the Prüfer domain is h-local, and in this case such factorizations have certain uniqueness properties. This leads to new characterizations of the h-local property in Prüfer domains. We also explore consequences of these factorizations and give illustrative examples.     

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial.     

Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C^2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C¹(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations.     

Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein–Lazarsfeld–Smith and Hochster–Huneke raised the containment problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci–Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero-dimensional subschemes of projective space. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals I$I$ of smooth subschemes, generalizing what is done in Bocci and Harbourne (2010)  [5]. We apply these bounds to ideals of unions of general lines in P^N${\mathbb{P}}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in P³${\mathbb{P}}^3$.     

On quasi divisor theories and systems of valuations

We study the relationship between divisor theories and systems of valuations, and characterize monoids with quasi divisor theories of finite character by systems of essential valuations. Throughout, we avoid ideal theory but use divisor theoretical methods.