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Preprints from February 2004Conca-SidmanAldo Conca and Jessica Sidman have posted a new preprint, "Generic initial ideals of points and curves", to the arXiv. Math Subject Class: 13P10, 14H50 Comments: AMS Latex, 20 pages Abstract: 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. Posted on February 27, 2004
YanagawaKohji Yanagawa has posted a new preprint, "BGG correspondence and Roemer's theorem on an exterior algebra", to the arXiv. Math Subject Class: 13D07; 18E30 Comments: 11 pages. To appear in Algebras and Representation Theory Abstract: Let E = K< y_1, ..., y_n > be the exterior algebra. The ``(cohomological) distinguished pairs" of a graded E-module M describe the growth of a minimal graded injective resolution of M. Roemer gave a duality theorem between the distinguished pairs of M and those of its dual M^*. In this paper, we show that under Bernstein-Gel'fand-Gel'fand correspondence, his theorem is translated into a natural corollary of local duality for (complexes of) graded S=K[x_1, >..., x_n]-modules. Using this idea, we also give a Z^n-graded version of Roemer's theorem. Posted on February 27, 2004
GreenDavid J. Green has posted a new preprint, "The essential ideal is a Cohen-Macaulay module", to the arXiv.
Comments: 6 pages Abstract: Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen-Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in the affirmative a question posed by J. F. Carlson. Posted on February 27, 2004
BöhningChristian Böhning has posted a new preprint, "L. Szpiro's conjecture on Gorenstein algebras in codimension 2", to the arXiv. Math Subject Class: 13D02 Comments: 20 pages Abstract: A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that R=Ext^2(R,A) holds as R-modules, A being a Cohen-Macaulay local ring with dim(A)-dim_A(R)=2. I prove a structure theorem for these algebras improving on an old theorem of M. Grassi. Special attention is paid to the question how the ring structure of R is encoded in its Hilbert resolution. It is shown that R is automatically a ring once one imposes a weak depth condition on a determinantal ideal derived from a presentation matrix of R over A. The interplay of Gorenstein algebras and Koszul modules as introduced by M. Grassi is clarified. Questions of applicability to canonical surfaces in P^4 have served as a guideline in these investigations. Posted on February 24, 2004
GuoLi Guo has posted a new preprint, "Baxter Algebras, Stirling Numbers and Partitions", to the arXiv. Math Subject Class: 16W99,11B73,05A18 Comments: 10 pages Abstract: Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial coefficients. This allows us to apply congruences from number theory to obtain congruences in Baxter algebras. Posted on February 24, 2004
Dibaei-YassemiMohammad T. Dibaei and Siamak Yassemi have posted a new preprint, "Generalized local cohomology and the Intersection Theorem", to the arXiv. Math Subject Class: 13D45, 13D22, 13D25, 13D05 Comments: 13 pages Abstract: Let $R$ be commutative Noetherian ring and let $\fa$ be an ideal of $R$. For complexes $X$ and $Y$ of $R$--modules we investigate the invariant $\inf{\mathbf R}\Gamma_{\fa}({\mathbf R}\Hom_R(X,Y))$ in certain cases. It is shown that, for bounded complexes $X$ and $Y$ with finite homology, $\dim Y\le\dim{\mathbf R}\Hom_R(X,Y)\le\pd X+\dim(X\otimes^{\mathbf L}_RY)+\sup X$ which strengthen the Intersection Theorem. Here $\inf X$ and $\sup X$ denote the homological infimum, and supremum of the complex $X$, respectively. Posted on February 24, 2004
BöhningChristian Böhning has posted a new preprint, "Canonical surfaces in P^4 and Gorenstein algebras in codimension 2", to the arXiv. Math Subject Class: 14J29 Comments: 40 pages Abstract: In this paper I investigate minimal surfaces of general type with p_g=5, q=0 for which the 1-canonical map is a birational morphism onto a surface in P^4 (so called canonical surfaces in P^4) via a structure theorem for the Hilbert resolutions of the canonical rings of the afore-mentioned surfaces, viewed as Gorenstein algebras of codimension 2 over the homogeneous coordinate ring of P^4. I discuss how the ring structure of such an algebra is encoded in its resolution. Among other things I show how this method can be applied to analyze the moduli space of canonical surfaces with p_g=5, q=0, K^2=11, thus recovering a result previously obtained by D. Rossberg with different techniques. Posted on February 24, 2004
TrivediV. Trivedi has posted a new preprint, "Semistability and Hilbert-Kunz multiplicities for curves", to the arXiv. Math Subject Class: 13D40 Comments: Latex2e, 14 pages Abstract: We study Hilbert-Kunz multiplicity of non-singular curves in positive characteristic. We analyse the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the HK multiplicity. This leads to a lower bound, achieved if the kernel sheaf is Frobenius semistable, and otherwise to formulas for the HK multiplicity in terms of parameters measuring the failure of Frobenius semistability. As a byproduct, an explicit example of a vector bundle on a curve is given whose $n$-th iterated Frobenius pullback is not semistable, while its $(n-2)$-nd such pullback is semistable, where $n>1$ is arbitrary. Posted on February 18, 2004
MustataMircea Mustata has posted a new preprint, "Multiplier ideals of hyperplane arrangements", to the arXiv. Math Subject Class: 14B05; 52C35 Comments: 11 pages Abstract: We compute the multiplier ideals of hyperplane arrangements via the interpretation of these ideals in terms of spaces of arcs, due to Ein, Lazarsfeld and the author. Posted on February 18, 2004
JorgensenPeter Jorgensen has posted a new preprint, "Tate cohomology over fairly general rings", to the arXiv. Math Subject Class: 13D02, 16E05, 18G25, 20J06 Comments: 15 pages Abstract: Tate cohomology was originally defined over finite groups. More recently, Avramov and Martsinkovsky showed how to extend the definition so that it now works well over Gorenstein rings. This paper improves the theory further by giving a new definition that works over more general rings, specifically, those with a dualizing complex. The new definition of Tate cohomology retains the desirable properties established by Avramov and Martsinkovsky. Notably, there is a long exact sequence connecting it to ordinary Ext groups. Posted on February 12, 2004
Brenner2Holger Brenner has posted a new preprint, "The rationality of the Hilbert-Kunz multiplicity in graded dimension two", to the arXiv. Math Subject Class: 13A35; 13D02; 13D40; 14H60 Abstract: We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. Moreover we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle\Syz(f_1, . . ., f_n) on the projective curve Y = Proj R. Posted on February 12, 2004
Brenner1Holger Brenner has posted a new preprint, "Tight closure and plus closure in dimension two", to the arXiv. Math Subject Class: 13A35; 14H60 Abstract: We prove that the tight closure and the graded plus closure of a homogeneous ideal coincide for a two-dimensional N-graded domain of finite type over the algebraic closure of a finite field. This answers in this case a ``tantalizing question'' of Hochster. Posted on February 12, 2004
Sturmfels-SullivantBernd Sturmfels and Seth Sullivant have posted a new preprint, "Toric ideals of phylogenetic invariants", to the arXiv. Comments: 28 pages, 3 figures Abstract: Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine generators and Gröbner bases for these toric ideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gröbner bases consist of certain explicitly constructed polynomials of degree at most four. Posted on February 11, 2004
GiorgiErika Giorgi has posted a new preprint, "On the irreducible components of the form ring and an application to intersection cycles", to the arXiv. Math Subject Class: 13B22 (Primary); 14C17 (Secondary) Comments: 14 pages Abstract: Let $A$ be a commutative noetherian ring and $I$ an ideal in $A$. We characterize algebraically when all the minimal primes of the associated graded ring $G_I A$ contract to minimal primes of $A/I$. This, applied to intersection theory, means that there are no embedded distinguished varieties of intersection. The characterization is in terms of the analytic spread of certain localizations of $I$, the symbolic Rees algebra and the normalization of the Rees algebra, and extends results of Huneke, Vasconcelos and Martí-Farré. Posted on February 10, 2004
Lassalle-SchlosserMichel Lassalle and Michael Schlosser have posted a new preprint, "Inversion of the Pieri formula for Macdonald polynomials", to the arXiv. Math Subject Class: 33D52 (Primary) 05E05, 15A09 (Secondary) Comments: 34 pages Abstract: We give the explicit analytic development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall-Littlewood symmetric functions. Posted on February 10, 2004
Lauritzen-Raben-Pedersen-ThomsenNiels Lauritzen, Ulf Raben-Pedersen, and Jesper Funch Thomsen have posted a new preprint, "Global F-regularity of Schubert varieties with applications to D-modules", to the arXiv. Math Subject Class: 32C38; 14B15 Abstract: We prove that Schubert varieties are globally F-regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic D-modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple D-modules coincide with local cohomology sheaves with support in Schubert varieties. Using a local Grothendieck-Cousin complex we prove that the decomposition of local cohomology sheaves with support in Schubert cells is multiplicity free. Posted on February 7, 2004
Matusevich-MillerLaura Matusevich and Ezra Miller have posted a new preprint, Combinatorics of rank jumps in simplicial hypergeometric systems, to the arXiv. Math Subject Class: 33C70 (Primary) 14M25, 13N10, 13D45, 52B20, 13C14, 16S36, 20M25 (Secondary) Abstract: Let A be an integer (d x n) matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d-1. Write \NA for the semigroup generated by the columns of A. It was proved by M. Saito [math.AG/0012257] that the semigroup ring \CC[\NA] over the complex numbers \CC is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system H_A(beta) equals the normalized volume of conv(A) for all complex parameters beta in \CC^d. Our refinement here shows, in this simplicial case, that H_A(beta) has rank strictly larger than the volume of conv(A) if and only if beta lies in the Zariski closure in \CC^d of all \ZZ^d-graded degrees where the local cohomology H^i_m(\CC[\NA]) at the maximal ideal m is nonzero for some i < d. Posted on February 6, 2004
OdaSusumu Oda has posted a new preprint, Endomorphisms of polynomial rings and Jacobians, to the arXiv. Math Subject Classes: 12C20;13F99 Abstract: The Jacobian Conjecture is established : If $f_1, ..., f_n$ be elements in a polynomial ring $k[X_1, ..., X_n]$ over a field $k$ of characteristic zero such that $ \det(\partial f_i/ \partial X_j) $ is a nonzero constant, then $k[f_1,..., f_n] = k[X_1, ..., X_n]$. Posted on February 5, 2004
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