Steven L. Kleiman

Introduction to Grothendieck duality theory

Preface.- Study of ?X.- Completions, primary decomposition and length.- Depth and dimension.- Duality theorems.- Flat morphisms.- Etale morphisms.- Smooth morphisms.- Curves.

Fundamental algebraic geometry : Grothendieck's FGA explained

Grothendieck topologies, fibered categories and descent theory: Introduction Preliminary notions Contravariant functors Fibered categories Stacks Construction of Hilbert and Quot schemes: Construction of Hilbert and Quot schemes Local properties and Hilbert schemes of points: Introduction Elementary deformation theory Hilbert schemes of points Grothendiecks existence theorem in formal geometry with a letter of Jean-Pierre Serre: Grothendiecks existence theorem in formal geometry The Picard scheme: The Picard scheme Bibliography Index.

Bertini theorems for hypersurface sections containing a subscheme

(1979). Bertini theorems for hypersurface sections containing a subscheme. Communications in Algebra: Vol. 7, No. 8, pp. 775-790.

Rational curves of degree at most 9 on a general quintic threefold

ABSTRACT. We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P 4. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (−1) ⊕ (−1), and in P 4 with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.

Autoduality of the Compactified Jacobian

The following autoduality theorem is proved for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1, form the corresponding Abel map AL:C→, which maps C into its compactified Jacobian, and form its pullback map , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, points of multiplicity 2, then is an isomorphism, and forming it commutes with specializing C. Much of the work in the paper is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. In this case, the determinant of cohomology is used to construct a right inverse to . Then a scheme-theoretic version of the theorem of the cube is proved, generalizing Mumfords, and it is used to prove that is independent of the choice of L. Finally, the autoduality theorem is proved. The presentation scheme is used to achieve an induction on the difference between the arithmetic and geometric genera; here, special properties of points of multiplicity 2 are used.

Bounding Solutions of Pfaff Equations

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.

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 1-bundle. Finally, using Abel maps of bidegree (m, 1), we characterize the curves having double points at worst.

The presentation functor and the compactified Jacobian

This article continues the development, begun in [2], [4], [3], [5] and [14], of a theory of the compactified Jacobian that is modeled and based on Grothendieck’s theory of the Picard scheme. Here we study the presentation functor, whose function is to bridge the gap between the compactified Jacobian J of a (relative) singular curve C and that J’ of one of its partial normalizations C’. We shall place special emphasis on the case in which each geometric fiber of C has just one singularity more than C’ and it is an ordinary node or cusp.

Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those with a Hilbert--Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade 1 can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade 1 that are birational onto their image, on the one hand, and self-linked perfect ideals of grade 2 that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.

Multiple Point Formulas for Maps

Let f: X → Y be a map. Its set of r-fold points is