arXiv subjects

S. Kleiman

Publications and source records attributed to S. Kleiman.

7 recordsLinked to original sources

arXiv2003

Bounds on leaves of foliations of the plane

This paper contributes to the solution of the Poincare problem, which is to bound the degree of a (generalized algebraic) leaf of a (singular algebraic) foliation of the complex projective plane. The first theorem gives a new sort of bound, which involves the Castelnuovo--Mumford regularity of the singular locus of the leaf. The second theorem gives a bound in terms of two singularity numbers of the leaf: the total Tjurina number, and the number of non-quasi-homogeneous singularities. If such singularities are present, then this bound improves one due to du Plessis and Wall, at least when the curve is irreducible.

math.AG↗

arXiv2003

Bounding solutions of Pfaff equations

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

math.AG↗

arXiv2002

Bounds on leaves of one-dimensional foliations

Let X be a variety over an algebraically closed field, η:Ω^1_X\to L a one-dimensional singular foliation, and C\subseteq X a projective leaf of η. We prove that 2p_a(C)-2=°(L|C)+λ(C)-°(C\cap S) where p_a(C) is the arithmetic genus, where λ(C) is the colength in the dualizing sheaf of the subsheaf generated by the Kähler differentials, and where S is the singular locus of η. We bound λ(C) and °(C\cap S), and then improve and extend some recent results of Campillo, Carnicer, and de la Fuente, and of du Plessis and Wall.

math.AG↗

arXiv2001

Node polynomials for families: results and examples

We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm Göttsche's conjecture about plane curves of low degree. Second, we justify Vainsencher's enumeration of irreducible six-nodal plane curves on a general quintic threefold in four-space. Third, we supplement Bryan and Leung's enumeration of nodal curves in a given homology class on an Abelian surface of Picard number one.

math.AG↗

arXiv2000

Abel Maps and Presentation Schemes

We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations provided by the images of certain Abel maps and certain presentation schemes. Third we study a lifting of the Abel map of bidegree (m,1) to the corresponding presentation scheme. Fourth we prove that, if a curve is blown up at a double point, then the corresponding presentation scheme is a IP^1-bundle. Finally, using Abel maps of bidegree (m,1), we characterize the curves having double points at worst

math.AG↗

arXiv1998

Conormal Geometry of Maximal Minors

Let A be a Noetherian local domain, N be a finitely generated torsion- free module, and M a proper submodule that is generically equal to N. Let A[N] be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A[M] be the subalgebra generated by M. Set C:=Proj(A[M]) and r:=dim C. Form the (closed) subset W of Spec(A) of primes p where A[N]_p is not a finitely generated module over A[M]_p, and denote the preimage of W in C by E. We prove this: (1) dim E=r-1 if either (a) N is free and A[N] is the symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2) E is equidimensional if (a) holds and A is universally catenary. Our proof was inspired by some recent work of Gaffney and Massey, which we sketch; they proved (2) when A is the ring of germs of a complex- analytic variety, and applied it to perfect a characterization of Thom's A_f-condition in equisingularity theory. From (1), we recover, with new proofs, the usual height inequality for maximal minors and an extension of it obtained by the authors in 1992. From the latter, we recover the authors' generalization to modules of B"oger's criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by the second author, to the geometry of the dual variety of a projective variety, and use it to obtain an interesting example where the conclusion of (1) fails and A[N] is a finitely generated module over A[M].

alg-geom↗

arXiv1998

Specialization of integral dependence for modules

We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as part of the general algebraic-geometric theory of the Buchsbaum--Rim multiplicity. Then we apply the principle to the study of equisingularity of ICIS germs, obtaining results for such equisingularity conditions as Whitney's Condition A, Thom's Condition A_f, and Henry, Merle and Sabbah's Condition W_f. Notably, we describe these conditions for analytic families in terms of various numerical invariants, which, for the most part, depend only on the members of a family, not on its total space.

alg-geom↗