Communications in Algebra

Aluffi, Paolo

doi: 10.1081/AGB-120022435pmid: N/A

Abstract We propose a variation of the notion of Segre class, by forcing a naive ‘inclusion-exclusion’ principle to hold. The resulting class is computationally tractable, and is closely related to Chern-Schwartz-MacPherson classes. We deduce several general properties of the new class from this relation, and obtain an expression for the Milnor class of an arbitrary scheme in terms of this class. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Caporaso, Lucia; Casagrande, Cinzia

doi: 10.1081/AGB-120022437pmid: N/A

Abstract The geometry of the moduli space of stable spin curves is studied, with emphasis on its combinatorial properties. In this context, the standard graph-theoretic framework is not just a book-keeping device: some purely combinatorial results are proved, having moduli- theoretic applications. In particular, certain strata of the moduli space of stable curves are characterized by a (finite) set of integers, measuring the non-reducedness of the scheme of spin curves, and definable in purely graph-theoretical terms. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Clemens, Herbert

doi: 10.1081/AGB-120022438pmid: N/A

Abstract A second-order invariant of C. Voisin gives a powerful method for bounding from below the geometric genus of a k-dimensional subvariety of a degree dhypersurface in complex projective n-space. This work uses the Voisin method to establish a general bound, which lies behind recent results of G. Pacienza and Z. Ran. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Corso, Alberto; Ghezzi, Laura; Polini, Claudia; Ulrich, Bernd

doi: 10.1081/AGB-120022439pmid: N/A

Abstract Let (R, 𝔪) be a Noetherian local ring and let Ibe an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ℱ = ℛ/𝔪ℛ of I, where ℛ denotes the Rees algebra of I. Our key idea is to require ‘good’ intersection properties as well as ‘few’ homogeneous generating relations in low degrees. In particular, if Iis a strongly Cohen-Macaulay R-ideal with G ℓand the expected reduction number, we conclude that ℱ is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of ℛ/Kℛ for any 𝔪-primary ideal K. This result recovers a well-known criterion of Valabrega and Valla whenever K = I. Furthermore, we study the relationship between the Cohen-Macaulay property of the special fiber ring ℱ and the Cohen-Macaulay property of the Rees algebra ℛ and the associated graded ring 𝒢 of I. Finally, we focus on the integral closedness of 𝔪I. The latter question is motivated by the theory of evolutions. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Edidin, Dan; Graham, William

doi: 10.1081/AGB-120022440pmid: N/A

Abstract In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne (Brion M. and Vergne M. ([1997]). An equivariant Riemann-Roch theorem for complete simplicial toric varieties. J. Reine. Agnew. Math.482:67–92) using different techniques. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Esteves, Eduardo; Gatto, Letterio

doi: 10.1081/AGB-120022441pmid: N/A

Abstract In the 1980's Cornalba and Harris discovered a relation among the Hodge class and the boundary classes in the Picard group with rational coefficients of the moduli space of stable, hyperelliptic curves. They proved the relation by computing degrees of the classes involved for suitable one-parameter families. In the present article we show that their relation can be obtained as the class of an appropriate, geometrically meaningful empty set, thus conforming with Faber's general philosophy of finding relations among tautological classes in the Chow ring of the moduli space of curves. The empty set we consider is the closure of the locus of smooth, hyperelliptic curves having a special ramification point.

Esteves, Eduardo; Kleiman, Steven L.

doi: 10.1081/AGB-120022442pmid: N/A

Abstract Let ω be a Pfaff system of differential forms on . Let Sbe its singular locus, and Ya solution of ω = 0. We prove Y ∩ Sis of codimension at most 1 in Y, just as Jouanolou suspected; he proved this result assuming ω is completely integrable, and asked if the integrability is, in fact, needed. Furthermore, we prove a lower bound on the Castelnuovo–Mumford regularity of Y ∩ S. As in two related articles, we derive upper bounds on numerical invariants of Y, thus contributing to the solution of the Poincaré problem. We work with Pfaff fields not necessarily induced by Pfaff systems, with ambient spaces more general than , and usually in arbitrary characteristic.

Franco, Davide

doi: 10.1081/AGB-120022443pmid: N/A

Abstract This paper is devoted to the study of some properties of a natural period mapassociated to SU(2) theta functions or, in other words, to SU(2) conformal vacua. In particular we will consider for such a map an analogue of the classical Torelli's problem. The main result of the paper is 5.17 where we give sufficient conditions for our period map having the infinitesimal Torelli property. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Gaffney, Terence

doi: 10.1081/AGB-120022444pmid: N/A

Abstract Given M, a submodule of Nwhich is in turn a submodule of a free R-module, where Ris a Noetherian local ring which is equidimensional and universally catenary, the theorem of Rees asserts that the integral closures of Mand Nagree provided their multiplicities agree. We extend this result to modules of non-finite colength by introducing a chain of submodules H i (M)which give an increasingly good approximation to the integral closure of M, and a sequence of numbers based on the multiplicity of a pair of modules. Some applications to problems in equisingularity are given. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

González, Sonia; Mallavibarrena, Raquel

doi: 10.1081/AGB-120022445pmid: N/A

Abstract The main objects of this paper are osculating spaces of order mto smooth algebraic curves, with the property of meeting the curve again. We prove that the only irreducible curves with an infinite number of this type of osculating spaces of order mare curves in P m+1whose degree nis greater than m + 1. This is a generalization of the result and proof of Kaji (Kaji, H. (1986). On the tangentially degenerate curves. J. London Math. Soc.33(2):430–440) that corresponds to the case m = 1. We also obtain an enumerative formula for the number of those osculating spaces to curves in P m+2. The case m = 1 of it is a classical formula proved with modern techniques by Le Barz (Le Barz, P. (1982). Formules multisécantes pour les courbes gauches quelconques. In: Enumerative Geometry and Classical Algebraic Geometry. Prog. in Mathematics 24, Birkhäuser, pp. 165–197). Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.