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Preprints from December 2003Takagi-WatanabeShunsuke Takagi and Kei-ichi Watanabe have posted a new preprint, "On F-pure thresholds", to the arXiv. Math Subject Class: 13A35; 14B05 Comments: 18 pages, AMS-LaTeX Abstract: Using the Frobenius map, we introduce a new invariant of a pair $(R,\a)$ of a ring $R$ and an ideal $\a \subset R$, which we call the F-pure threshold $\mathrm{c}(\a)$ of $\a$, and study its properties. We see that the F-pure threshold characterizes some ring theoretic properties. By virtue of Hara and Yoshida's result \cite{HY}, the F-pure threshold $\mathrm{c}(\a)$ in characteristic zero corresponds to the log canonical threshold $\mathrm{lc}(\a)$ which is an important invariant in birational geometry. Using F-pure thresholds, we prove some ring theoretic properties for three-dimensional terminal singularities of characteristic zero. Also, in fixed prime characteristic, we establish several properties of F-pure thresholds similar to those of log canonical thresholds with quite simple proofs. Posted on December 31, 2003
Terai-YoshidaNaoki Terai and Ken-ichi Yoshida have posted a new preprint, "Buchsbaum Stanley--Reisner rings with minimal multiplicity", to the arXiv. Math Subject Class: 13F55; 13H10; 13D02 Comments: about 25 pages, LaTeX Abstract: In this paper, we study non-Cohen--Macaulay Buchsbaum Stanley--Reisner rings with linear free resolution. In particular, for given integers $c$, $d$, $q$ with $c \ge 1$, $2 \le q \le d$, we give an upper bound $h_{c,d,q}$ on the dimension of the unique non-vanishing homology $\widetilde{H}_{q-2}(\Delta;k)$ of a $d$-dimensional Buchsbaum ring $k[\Delta]$ with $q$-linear resolution and codimension $c$. Also, we discuss about existence for such Buchsbaum rings with $\dim_k \widetilde{H}_{q-2}(\Delta;k) = h$ for any $h$ with $0 \le h \le h_{c,d,q}$, and prove an existence theorem in the case of $q=d=3$ using the notion of Cohen--Macaulay linear cover. On the other hand, we introduce the notion of Buchsbaum Stanley--Reisner rings with minimal multiplicity of type $q$, which extends the notion of Buchsbaum rings with minimal multiplicity defined by Goto. As an application, we give many examples of Buchsbaum Stanley--Reisner rings with $q$-linear resolution. Posted on December 31, 2003
WelkerPatricia Hersh Volkmar Welker has posted a new preprint, "Groebner basis degree bounds on $\Tor^{k[\Lambda ]}_\bullet(k,k)_\bullet$ and discrete Morse theory for posets", to the arXiv. Math Subject Class: 05E25; 06A07; 13P10; 13D02 Abstract: The purpose of this paper is twofold. 1. We give combinatorial bounds on the ranks of the groups $\Tor^{R}_\bullet(k,k)_\bullet$ in the case where $R = k[\Lambda]$ is an affine semi-group ring, and in the process provide combinatorial proofs for bounds by Eisenbud, Reeves and Totaro on which Tor groups vanish. In addition, we show that if the bounds hold for a field $k$ then they hold for $\field[\Lambda]$ and any field $\field$. Moreover, we provide a combinatorial construction for a free resolution of $\field$ over $\field[\Lambda]$ which achieves these bounds. 2. We extend the lexicographic discrete Morse function construction of Babson and Hersh for the determination of the homotopy type and homology of order complexes of posets to a larger class of facet orderings that includes orders induced by monomial term orders. Since it is known that the order complexes of finite intervals in the poset of monomials in $k[\Lambda]$ ordered by divisibility in $k[\Lambda ]$ govern the $\Tor$-groups, the newly developed tools are applicable and serve as the main ingredients for the proof of the bounds and the construction of the resolution. Posted on December 31, 2003
TuLoring W. Tu has posted a new preprint, "A generalized Vandermonde determinant", to the arXiv. Math Subject Class: Primary: 15A15, 41A10; Secondary: 05A17, 06A06 Comments: 5 pages; to appear in the Journal of Algebra Abstract: Interpolation theory suggests a generalization of the usual Vandermonde determinant to numbers with multiplicities. We prove a discriminant formula for this generalized Vandermonde determinant and give an application to the Hermite interpolation problem. Posted on December 25, 2003
Avramov-Iyengar-MillerLuchezar L. Avramov, Srikanth Iyengar, and Claudia Miller have posted a new preprint, "Homology over local homomorphisms", to the arXiv. Math Subject Class: 13D05, 13D40, 13H10 Comments: 52 pages Abstract: The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R-modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R. A third is to obtain criteria for a ring equipped with a `contracting' endomorphism -- such as the Frobenius endomorphism -- to be regular or complete intersection; these results represent broad generalizations of Kunz's characterization of regularity in prime characteristic. Posted on December 25, 2003
FrenkelP. E. Frenkel has posted a new preprint, "Simple proof of Chebotarev's theorem", to the arXiv. Math Subject Class: 11T22 Comments: 2 pages Abstract: We give a simple proof of Chebotarev's theorem: Let $p$ be a prime and $\omega $ a primitive $p$th root of unity. Then all minors of the matrix $(\omega^{ij})_{i,j=0}^{p-1}$ are non-zero. Posted on December 25, 2003
Luengo-Velasco-Melle-Hernandez-NemethiI. Luengo-Velasco, A. Melle-Hernandez, and A. Nemethi have posted a new preprint, "Links and analytic invariants of superisolated singularities", to the arXiv. Math Subject Class: 14B05,14J17,32S25,57M27,57R57 Abstract: By using superisolated surface singularities whose link is a rational homology sphere we give counterexamples to some of the most important conjetures concernig invariants of normal surface singularities. Posted on December 25, 2003
D'AndreaCarlos D'Andrea has posted a new preprint, "On the irreducibility of the determinant of the matrix of moving planes", to the arXiv. Math Subject Class: 14Qxx; 13Pxx Comments: 9 pages, amsart Abstract: We show that, for generic bihomogeneous polynomials, the determinant of the matrix of moving planes is irreducible. Posted on December 22, 2003
Buhler-ReichsteinJoe Buhler and Zinovy Reichstein have posted a new preprint, "Symmetric functions and the phase problem in crystallography", to the arXiv. Math Subject Class: 05E05, 13A50, 13P99, 20C10 Comments: 28 pages, to appear in Transactions AMS Abstract: The calculation of crystal structure from X-ray diffraction data requires that the phases of the ``structure factors'' (Fourier coefficients) determined by scattering be deduced from the absolute values of those structure factors. Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case of crystal structures with $n$ equal atoms in the unit cell, with $n$ small. We rephrase the problem as a question about multiplicative invariants for a particular finite group action. We show that the absolute values form a generating set for the field of invariants of this action, and consider the problem of making this theorem constructive and practical; the most promising approach for deriving explicit formulas uses SAGBI bases. Posted on December 22, 2003
KatzmanMordechai Katzman has posted a new preprint, "The support of top graded local cohomology modules", to the arXiv. Abstract: Let $R_0$ be any domain, let $R=R_0[U_1, ..., U_s]/I$, where $U_1, ..., U_s$ are indeterminates of some positive degrees, and $I\subset R_0[U_1, ..., U_s]$ is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of $H:=H^s_{R_+}(R)$ contain a certain non-zero ideal $c(I)$ of $R_0$ called the ``content'' of $I$. It follows that the support of $H$ is simply $V(\content(I)R + R_+)$ (Corollary 1.8) and, in particular, $H$ vanishes if and only if $c(I)$ is the unit ideal. These results raise the question of whether local cohomology modules have finitely many minimal associated primes-- this paper provides further evidence in favour of such a result. Finally, we give a very short proof of a weak version of the monomial conjecture based on these results. Posted on December 18, 2003
LorenzMartin Lorenz has posted a new preprint, "On the Cohen-Macaulay property of multiplicative invariants", to the arXiv. Math Subject Class: 13A50; 16W22; 13C14; 13H10 Comments: 13 pages Abstract: We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group $G$. By definition, these are $G$-actions on Laurent polynomial algebras that stabilize the multiplicative group consisting of all monomials in the variables. For the most part, we concentrate on the case where the base ring is the ring of rational integers. Our main result states that if $G$ acts non-trivially and the invariant algebra is Cohen-Macaulay then the abelianized isotropy groups $G_m/[G_m,G_m]$ of all monomials m are generated by bireflections and at least one $G_m/[G_m,G_m]$ is non-trivial. As an application, we prove the multiplicative version of Kemper's 3-copies conjecture. Posted on December 18, 2003
Miro-Roig-RanestadR. M. Miro-Roig and K. Ranestad have posted a new preprint, "Intersection of ACM-curves in P^3", to the arXiv. Math Subject Class: 14C17 (Primary) 14H45 (Secondary) Comments: 15 pages Abstract: In this note we address the problem of determining the maximum number of points of intersection of two arithmetically Cohen-Macaulay curves in $\PP^3$. We give a sharp upper bound for the maximum number of points of intersection of two irreducible arithmetically Cohen-Macaulay curves $C_t$ and $C_{t-r}$ in $\PP^3$ defined by the maximal minors of a $t \times (t+1)$, resp. $(t-r) \times (t-r+1)$, matrix with linear entries, provided $C_{t-r}$ has no linear series of degree $d\leq{{t-r+1}\choose 3}$ and dimension $n\geq t-r$. Posted on December 18, 2003
KarpenkoNikita A. Karpenko has posted a new preprint, "Holes in I^n", to the arXiv. Math Subject Class: 11E04; 14C25 Comments: 29 pages Abstract: Let F be an arbitrary field of characteristic not 2. We write W(F) for the Witt ring of F, consisting of the isomorphism classes of all anisotropic quadratic forms over F. For any element x of W(F), dimension dim x is defined as the dimension of a quadratic form representing x. The elements of all even dimensions form an ideal denoted I(F). The filtration of the ring W(F) by the powers I(F)^n of this ideal plays a fundamental role in the algebraic theory of quadratic forms. The Milnor conjectures, recently proved by Voevodsky and Orlov-Vishik-Voevodsky, describe the successive quotients I(F)^n/I(F)^{n+1} of this filtration, identifying them with Galois cohomology groups and with the Milnor K-groups modulo 2 of the field F. In the present article we give a complete answer to a different old-standing question concerning I(F)^n, asking about the possible values of dim x for x in I(F)^n. More precisely, for any positive integer n, we prove that the set dim I^n of all dim x for all x in I(F)^n and all F consisists of 2^{n+1}-2^i, i=1,2,...,n+1 together with all even integers greater or equal to 2^{n+1}. Previously available partial informations on dim I^n include the classical Arason-Pfister theorem, saying that no integer between 0 and 2^n lies in dim I^n, as well as a recent Vishik's theorem, saying the same on the integers between 2^n and 2^n+2^{n-1} (the case n=3 is due to Pfister, n=4 to Hoffmann). Our proof is based on computations in Chow groups of powers of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties. Posted on December 16, 2003
JorgensenPeter Jorgensen has posted a new preprint, "The Gorenstein projective modules are precovering", to the arXiv. Math Subject Class: 18G25, 16D40, 13C10 Comments: 11 pages Abstract: The Gorenstein projective modules are proved to form a precovering class in the module category of a ring which has a dualizing complex. Posted on December 16, 2003
NagelUwe Nagel has posted a new preprint, "Comparing Castelnuovo-Mumford regularity and extended degree: the borderline cases", to the arXiv. Abstract: Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, Vasconcelos shows that both can be compared in case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained if we choose as extended degree the homological or the smallest extended degree. The corresponding algebras are characterized in three ways: by relations among the algebra generators, by using generic initial ideals, and by their Hilbert series. Posted on December 4, 2003
Nagel-Notari-SpreaficoUwe Nagel, Roberto Notari, and Maria Luisa Spreafico have posted a new preprint, "The Hilbert scheme of degree two curves and certain ropes", to the arXiv. Abstract: We study families of ropes of any codimension that are supported on lines. In particular, this includes all non-reduced curves of degree two. We construct suitable smooth parameter spaces and conclude that all ropes of fixed degree and genus lie in the same component of the corresponding Hilbert scheme. We show that this component is generically smooth if the genus is small enough unless the characteristic of the ground field is two and the curves under consideration have degree two. In this case the component is non-reduced. Posted on December 4, 2003
Chiarli-Greco-NagelNadia Chiarli, Silvio Greco, and Uwe Nagel have posted a new preprint, "Normal form for space curves in a double plane", to the arXiv. Abstract: This note is an attempt to relate explicitly the geometric and algebraic properties of a space curve that is contained in some double plane. We show in particular that the minimal generators of the homogeneous ideal of such a curve can be written in a very specific form. As applications we characterize the possible Hartshorne-Rao modules of curves in a double plane and the minimal curves in their even Liaison classes. Posted on December 4, 2003
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