The Development of Intersection Homology Theory

This historical introduction is in two parts. The first is reprinted with permission from ``A century of mathematics in America, Part II,'' Hist. Math., 2, Amer. Math. Soc., 1989, pp.543-585. Virtually no change has been made to the original text. In particular, Section 8 is followed by the original list of references. However, the text has been supplemented by a series of endnotes, collected in the new Section 9 and followed by a second list of references. If a citation is made to the first list, then its reference number is simply enclosed in brackets -- for example, [36]. However, if a citation is made to the second list, then its number is followed by an `S' -- for example, [36S]. Further, if a subject in the reprint is elaborated on in an endnote, then the subject is flagged in the margin by the number of the corresponding endnote, and the endnote includes in its heading, between parentheses, the page number or numbers on which the subject appears in the reprint below. Finally, all cross-references appear as hypertext links in the dvi and pdf copies.Comment: 58 pages, hypertext links added; appeared in Part 3 of the special issue of Pure and Applied Mathematics Quarterly in honor of Robert MacPherson. However, the flags in the margin were unfortunately (and inexplicably) omitted from the published versio

Two Formulas for the BR Multiplicity

We prove a projection formula, expressing a relative Buchsbaum--Rim multiplicity in terms of corresponding ones over a module-finite algebra of pure degree, generalizing an old formula for the ordinary (Samuel) multiplicity. Our proof is simple in spirit: after the multiplicities are expressed as sums of intersection numbers, the desired formula results from two projection formulas, one for cycles and another for Chern classes. Similarly, but without using any projection formula, we prove an expansion formula, generalizing the additivity formula for the ordinary multiplicity, a case of the associativity formula.Comment: 10 pages, to appear in the Annali dell'Universit\`a di Ferrara, in a special memorial volume honoring Bobi Lascu. This version has been revised following a referee's suggestions, but the technical mathematics is unchange

Toward Clemens' Conjecture in degrees between 10 and 24

We introduce and study a likely condition that implies the following form of Clemens' conjecture in degrees $d$ between 10 and 24: given a general quintic threefold $F$ in complex \IP^4, the Hilbert scheme of rational, smooth and irreducible curves $C$ of degree $d$ on $F$ is finite, nonempty, and reduced; moreover, each $C$ is embedded in $F$ with balanced normal sheaf \O(-1)\oplus\O(-1), and in \IP^4 with maximal rank.Comment: Plain Tex, This eleven page paper is a joint manuscript, produced in connection with the first author's participation in the conference "Geometry and Physics", Zlatograd, Bulgaria, August 28 - Sept.2, 1995." This version contains a small change in Remark (3.3); the hope expressed there has been refine

The Picard Scheme

This article introduces, informally, the substance and the spirit of Grothendieck's theory of the Picard scheme, highlighting its elegant simplicity, natural generality, and ingenious originality against the larger historical record.Comment: To appear in "Alexandre Grothendieck: A Mathematical Portrait" edited by Leila Schneps and published by International Press of Bosto

Macaulay duality and its geometry

Macaulay Duality, between quotients of a polynomial ring over a field, annihilated by powers of the variables, and finitely generated submodules of the ring's graded dual, is generalized over any Noetherian ring, and used to provide isomorphisms between the subschemes of the Hilbert scheme parameterizing various sorts of these quotients, and the corresponding subschemes of the Quot scheme of the dual. Thus notably the locus of recursively compressed algebras of permissible socle type is proved to be covered by open subschemes, each one isomorphic to an open subscheme of a certain affine space. Moreover, the polynomial variables are weighted, the polynomial ring is replaced by a graded module, and attention is paid to induced filtrations and gradings. Furthermore, a similar theory is developed for (relatively) maximal quotients of a graded Gorenstein Artinian algebra.Comment: Clarifications were made following the many thoughtful suggestions of the referee of the Collino Memorial Volume, including new indices of terminology and notation and the observation (justified more in this version than the one in press) that a general quotient of socle type bounded by t(-) is RECURSIVELY compressed of socle type t(-