Research Abstracts

Much of my research in algebraic geometry has been motivated by the problem of finding sharp vanishing theorems for the higher cohomology of stable rank two bundles on three dimensional projective space over an arbitrary algebraically closed field. In attacking this from various directions, I've learned about linkage theory, vanishing theorems, cohomological bounding techniques and Hilbert schemes. Details can be found in the abstracts below. The reference numbers refer to the bibliography at the end.

Currently I've been trying to broaden the scope of my research. While I continue to think about the topics above, I've been recently interested in computing Picard groups of general (singular) surfaces which contain a general curve. Conversations with Frederico Xavier have resulted in some results in differential geometry which have applications to the Jacobian conjecture.

Even Linkage Classes [Trans. Amer. Math. Soc. 348 no. 3 (1996) 1137-1162.]

Most of the work in liaison theory deals with locally Cohen-Macaulay subschemes in projective space. However Hartshorne's theory of generalized divisors [16] shows that the foundations of this theory work just as well for subschemes without embedded points. The main result in this paper extends Rao's correspondence [42] to a bijection

We further give an explicit description of stable equivalence classes of reflexive sheaves: it turns out that each such class has a unique element E0 of minimal rank and each element of the class is obtained from E0 by adding direct summands of line bundles. This simple description is the key to showing that the main results of Martin-Deschamps and Perrin for space curves [25] generalize to higher dimension. In particular, we recover the structure theorem of Ballico, Bolondi and Migliore [2] in this more general setting.

Smooth Curves Linked to Thick Curves [Compositio Math. 106 no. 2 (1997) 203-210.]

A conjecture of Martin-Deschamps and Perrin [26] states that if a space curve C is a smooth and connected and D is linked to C by surfaces of minimal degree, then D has generic embedding dimension two. In this paper we show that if W is a smooth connected curve, then a general pair of surfaces S and T of high degree which contain the dth infinitesimal neighborhood Wd (for d > 1) link a locally Cohen-Macaulay multiplicity structure D supported on W to a smooth connected curve C. Since D must have local embedding dimension 3, this gives a counterexample to the conjecture. Surprisingly, it turns out that every rational quintic curve not lying on a quadric surface gives such a counterexample via a unique complete intersection of two cubic surfaces.

The Hilbert schemes of degree three curves [Ann. Sci. Ecole Norm. Sup. (4) 30 no. 3 (1997) 367-384.]

Here we completely classify the locally Cohen-Macaulay space curves C of degree three. From this we determine the number and dimensions of the irreducible components of the associated Hilbert schemes H(3,g). It turns out that there are roughly -g/3 components and that most of the components consist entirely of triple lines.

Perhaps the most interesting aspect of this paper deals with connectedness of the corresponding Hilbert schemes H(3,g). Working with explicit equations, we gave deformations which show that there are certain (extremal) triple lines which lie in the closure of each irreducible component, which shows that the Hilbert schemes H(3,g) of locally Cohen-Macaulay curves C of degree 3 and genus g are connected. This is one of the first examples of connectedness in such moduli spaces which does not follow from irreducibility, for example H(2,g) is irreducible and H(d,g) is irreducible for g large enough. It is an open question whether H(d,g) is connected in general. The deformations given here have found other connectedness applications in [18] and [36]; see also [14].

Subextremal curves [Manuscripta Math. 94 no. 3 (1997) 303-317.]

In the short and elegant paper [27], Martin-Deschamps and Perrin gave a sharp upper bound on the Rao function rC(n) = h1 IC(n) of a space curve C in terms of the degree d and the genus g of C. Their bound is sharp, and the curves which achieve their bound are called extremal curves. Exploiting a new invariant (called the spectrum for space curves developed in the Ph.D. thesis of Enrico Schlesinger [43], we improve known bounds of Ellia [9] on the Rao function of curves C which are not extremal. We show that the new bounds are sharp and classify the (sub)-extremal examples. The most difficult part involves handling certain subtleties which occur in characteristic p > 0. An analogous result holds for curves of degree d which contain a plane curve of degree d-p [38].

Degrees of generators of ideals [Comm. Algebra 26 no. 4 (1998) 1209-1231.]

In this paper we give an upper bound on the highest degree of a generator of the ideal of a locally Cohen-Macaulay projective scheme in terms of its Hilbert function. Specializing to space curves, we find that the curves for which the bound is sharp satisfy an interesting cohomological property. Moreover, we determine which even linkage classes possess such curves and show that this subset satisfies the Lazarsfeld-Rao property.

Bounds on c3 for threefolds [Manuscripta Math. 97 no. 2 (1998) 135-141.]

In [19], Hunt initiated a study of the geography of threefolds, which is basically the study of which Chern triples (c1,c2,c3) can occur for smooth threefolds satisfying certain conditions. Hunt explains that these conditions should be ampleness of the canonical bundle and minimality in the sense of the Mori program. The corresponding program for surfaces is relatively complete [39].

In the present work, we were able to bound the higher cohomology of the tangent and cotangent bundles of a smooth minimal threefold with ample canonical bundle to give a quadratic bound on the third Chern number in terms of the first two Chern numbers. There are better bounds in the case that one considers threefolds with principal Picard group generated by a very ample divisor [7].

Bounds on Multisecants [Collect. Math. 49 no. 2-3 (1998) 447-463.]

Here we recover a known upper bound [4] on the largest degree of a generator for the total ideal of an arithmetically Cohen-Macaulay subscheme V of codimension two in projective space (in terms of the Hilbert function) by a simple linking trick. This obviously gives an upper bound on the order of a multisecant line to V. The interesting part is the construction of sharp examples for the bound given.

We give an explicit description of the singular locus of the blow up of a regular local ring at a complete intersection ideal, which is the key to refining a standard linking theorem [40]. In particular, this improves on a construction of Gruson and Peskine for curves in P3 [13] in that (a) the characteristic zero hypothesis is removed, (b) the curves lie on smooth surfaces of minimal degree and (c) the curves have maximal order multisecant lines. The construction is similar to that of [23] except that the local algebra above shows that things work in finite characteristic. This method also recovers the consequence of Chang's filtered Bertini theorem [6] that a general arithmetically Cohen-Macaulay subscheme of codimension two has singular locus of codimension at most 4.

Hilbert polynomials over Artinian rings [Illinois J. Math. 43 no. 2 (1999) 338-349.]

Given an Artinian ring R and a homogeneous ideal I in S = R[x0,x1...xn], consider the associated Hilbert function H(n) = lR (S/I)n. In this paper we describe all such functions H arising in this way, extending the original results of Macaulay (see [11] for a modern treatment) in the case when R is a field. As a corollary we characterize the associated Hilbert polynomials. We also extend Gotzmann's regularity and persistence theorems to this setting.

Integral subschemes of codimension two [J. Pure Appl. Algebra 141 no. 3 (1999) 269-288.]

An even linkage class L of codimension two subschemes V in Pn is partially ordered by domination [25,30]. The main theorem of this paper gives a necessary condition for X in L to be an integral subscheme in terms of its location in the poset structure of L. After a suitable deformation, this condition is also sufficient for elements in L which dominate some integral subscheme. In particular, the conditions are both necessary and sufficient in the case originally studied by Lazarsfeld and Rao [20].

Bounds on the Rao function [J. Pure Appl. Algebra 152 no. 3 (2000) 253-266.]

Similar to "Subextremal Curves" above, the aim of this paper is to bound the Rao function rC(n) associated to a space curve C. However, in this work we take the minimal degree s of a surface containing the curve C into account. The extremal curves of Martin-Deschamps and Perrin [27,28] and the subextremal curves in [23] necessarily lie on degenerate surfaces of degree 2, so we are mainly interested in the case s > 2. The new idea here is to use a combinatorial object associated to the curve C (developed in the Ph.D. thesis of Rich Liebling [21]) called the triangle diagram, which arises naturally from the generic initial ideal. The bounds we obtain are sharp for s < 5.

On the second section of a rank two bundle [Comm. Algebra 28 no. 12 (2000) 5531-5540.]

For stable rank two reflexive sheaves E on P3, Hartshorne gave a sharp upper bound on the least number a such that E(a) must have a nonzero section in terms of the Chern classes of E [15]. In the present paper we give a similar bound (in terms of the Chern classes of E and a) for the least number b such that E(b) has two independent sections along with examples suggesting sharpness.

A remark on connectedness in Hilbert schemes [Comm. Algebra 28 no. 12 (2000) 5745-5747.]

In this short note we show that for any pair (d,g) for which the Hilbert scheme H(d,g) has subextremal curves, these subextemal curves can be deformed to extremal curves. As a corollary, we find that the Hilbert scheme H(d,g) is connected when g=1/2 (d-3)(d-4) +1. It has recently been shown in the Ph.D. thesis of Samir [1] that H(d,g) is connected when g=1/2 (d-3)(d-4), a more difficult case.

Global Inversion via the Palais-Smale condition [Disc. Cont. Dynam. Sys. - Ser. A 8 no. 1 (2002) 17-28.]

Fixing a complete Riemannian metric g on Rn, we show that a local diffeomorphism f:Rn -> Rn is bijective if the height function (standard inner product) satisfies the Palais-Smale condition relative to g for each for each nonzero v in Rn. Our method substantially improves a global inverse function theorem of Hadamard []. In the context of polynomial maps, we obtain new criteria for invertibility in terms of Lojasiewicz exponents and tameness of polynomials.

Curves on a double surface [To appear in Collectanea Math.]

In this short note we study families of curves which lie on the doubling of a smooth surface X inside a smooth 3-fold, following the method introduced in [17], where very satisfactory results were obtained for curves lying on a double plane 2H in P3. We show that a curve C on 2X in general gives rise to a triple (Z,R,P), where P is an effective Cartier divisor on X containing the effective Cartier divisor R, which in turn contains Z as a zero dimensional Gorenstein divisor.

The Hilbert schemes of degree four curves [To appear in Compositio Math.]

In this paper we describe the irreducible components of the Hilbert scheme of space curves of degree four and fixed arithmetic genus g in P3. Moreover, we produce deformations of the corresponding families of curves which shows that these Hilbert schemes are connected (it is still not known whether they are connected in general). Finally, we observe that the irreducible components W1 corresponding to extremal curves and W2 corresponding to thick multiplicity structures on a line do not intersect, thus producing a counterexample to a conjecture of Ait-amrane and Perrin [2].

Holomorphic injectivity and the Hopf map [Preprint, July 2003]

In this paper we ask under what conditions an etale morphism F: X -> Cn must be injective. When the codimension of the complement of the image is at least two, we find that F is injective if and only if each nonempty pre-image of a complex line is a connected rational curve. The method involves looking at the underlying complex manifolds and employs the classical Gronwall-Bieberbach estimates for functions on the unit disk as well as the fact that the Hopf map admits no continuous sections.

Genus bounds in arbitrary characteristic [Preprint, summer 2003]

In this paper we extend to finite characteristic two classical genus bounds of Halphen and Castelnuovo, proved in the modern language by Joe Harris []. This work was begun by Ballico [], who proved that the bound for curves of degree d > k(k-1) lying on an integral surface of degree k is valid in characteristic p>0; he also showed that Halphen's genus bound for integral curves not contained in a quadric holds for curves of degree d>25. Using the spectrum as an invariant, we are able to furnish a short uniform proof that works for all d>0.

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