arXiv subjects

Steven L. Kleiman

Publications and source records attributed to Steven L. Kleiman.

13 recordsLinked to original sources

arXiv2023

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.

math.AG↗

arXiv2019

On the Quot scheme $\mathrm{Quot}_{\mathcal O_{\mathbb P^1}^r/\mathbb P^1/k}^d$

We consider the quot scheme $\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k}$ of locally free quotients of $\mathcal F^r:= \bigoplus ^{ r} \mathcal O_{\mathbb P^1 }$ with Hilbert polynomial $p(t)=d$. We prove that it is a smooth variety of dimension $dr$, locally isomorphic to $\mathbb A^{dr}$. We introduce a new notion of support for modules in $\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k}$, called Hilb-support that allows us to define a natural surjective morphism of schemes $ξ:\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k} \to \mathrm{Hilb}^d_{\mathcal O_{\mathbb P^1}} $ associating to each module its Hilb-support and study the fibres of $ξ$ over each $k$-point $Z$ of $\mathrm{Hilb}^d_{\mathcal O_{\mathbb P^1}}$. If $Z=Y_1+\dots+Y_n$, with $Y_j=t_jR_j$, where $R_1, \dots, R_n$ are distinct points, the fibre of $ξ$ over $Z$ is isomorphic to $\mathrm{Quot}^{t_1}_{\mathcal F\otimes \mathcal O_{Y_1}/ Y_1/ k}\times\dots \times \mathrm{Quot}^{t_n}_{\mathcal F\otimes \mathcal O_{Y_n}/ Y_n/ k}$. We then study the Quot scheme $\mathrm{Quot}^{t}_{\mathcal F^r\otimes \mathcal O_{Y}/ Y/ k}$ with $Y=tR$. For $t=1$, $\mathrm{Quot}^{t}_{\mathcal F^r\otimes \mathcal O_{Y}/ Y/ k}$ is isomorphic to $\mathbb P^{r-1}$, while for $t\geq 2$ we prove that it is formed by a main irreducible, reduced and singular component of dimension $t(r-1)$ and by some embedded component of lower dimension.

math.AG↗

arXiv2016

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.

math.AG↗

arXiv2014

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.

math.HO↗

arXiv2013

On the Göttsche Threshold

For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample. For S rational, we relax the latter condition substantially: it suffices that three key loci be of codimension more than d. As corollaries, we prove that the condition conjectured by Göttsche suffices if S is P^2 or S is any Hirzebruch surface, and that a similar condition suffices if S is any classical del Pezzo surface.

math.AG↗

arXiv2013

Curve Counting à la Göttsche

Let n_δbe the number of δ-nodal curves lying in a suitably ample complete linear system |L| and passing through appropriately many points on a smooth projective complex algebraic surface. A major open problem is to understand the behavior of n_δ, specifically to finish off Lothar Göttsche's mostly proved 1997 conjectures and then go on to treat the new refinements by Göttsche and Vivek Shende. Five subproblems are explained, and work on them is surveyed.

math.AG↗

arXiv2008

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.

math.HO↗

arXiv2008

The Canonical Model of a Singular Curve

We give refined statements and modern proofs of Rosenlicht's results about the canonical model C' of an arbitrary complete integral curve C. Notably, we prove that C and C' are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C' is equal to the blowup of C with respect to the canonical sheaf ω. We also prove some new results: we determine just when C' is rational normal, arithmetically normal, projectively normal, and linearly normal.

math.AG↗

arXiv2005

The Picard scheme

We develop in detail most of the theory of the Picard scheme that Grothendieck sketched in two Bourbaki talks and in commentaries on them. Also, we review in brief much of the rest of the theory developed by Grothendieck and by others. But we begin with a twelve-page historical introduction, which traces the development of the ideas from Bernoulli to Grothendieck, and which may appeal to a wider audience.

math.AG↗

arXiv2000

Gherardelli linkage and complete intersections

Our main theorem characterizes the complete intersections of codimension 2 in a projective space of dimension 3 or more over an algebraically closed field of characteristic 0 as the subcanonical and self-linked subschemes. In order to prove this theorem, we'll prove the Gherardelli linkage theorem, which asserts that a partial intersection of two hypersurfaces is subcanonical if and only if its residual intersection is, scheme-theoretically, the intersection of the two hypersurfaces with a third.

math.AG↗

arXiv1997

Bertini and his two fundamental theorems

After reviewing Bertini's life story, a fascinating drama, we make a critical examination of the old statements and proofs of Bertini's two fundamental theorems, the theorem on variable singular points and the theorem on reducible linear systems. We explain the content of the statements in a way that is accessible to a nonspecialist, and we develop versions of the old proofs that are complete and rigorous by current standards. In particular, we prove a new extension of Bertini's first theorem, which treats variable $r$-fold points for any $r$.

alg-geom↗

arXiv1996

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 $Ø(-1)\oplusØ(-1)$, and in $\IP^4$ with maximal rank.

alg-geom↗

arXiv1996

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

We prove the following form of the Clemens conjecture in low degree. Let $d\le9$, and let $F$ be a general quintic threefold in $\IP^4$. 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 $Ø(-1)\oplusØ(-1)$, and in $\IP^4$ 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.

alg-geom↗