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 - this is joint work with John Brevik at Cal State Long Beach. Conversations with Frederico Xavier at Notre Dame 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.]
Previously liaison theory dealt exclusively with locally Cohen-Macaulay subvarieties of codimension two, but this is very restrictive: for example the cone C(V) over a variety V is not locally Cohen-Macaulay at its vertex, unless V happens to be arithmetically Cohen-Macaulay. Since the definitions work well for equidimensional subvarieties, this is the natural setting for liaison theory. In this paper I show that all the standard results carry over to the equidimensional case. In particular, we extend Rao's correspondence [Invent. Math. 50 (1979), Math. Ann. 258 (1981)] to a bijection
Smooth Curves Linked to Thick Curves [Compositio Math. 106 no. 2 (1997) 203-210.]
A conjecture of Martin-Deschamps and Perrin [Construction de Courbes Lisses: un Théoème de Bertini, Laboratoire de Mathematiques de l'École Normale Supereure 22 (1992)] 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.]
Hartshorne proved in his thesis that the full Hilbert scheme is connected [I.H.E.S. Publ. Math 29 (1966)], however connectedness for Hilbert schemes of space curves without embedded or isolated points remains open. It is well-known that such Hilbert schemes of degree two curves is connected, because it is irreducible, as follows from the classification of Migliore [On linking double lines, Trans. Amer. Math. Soc. 294 (1986)]. In this paper I gave the first example of connectedness in such a Hilbert scheme with many irreducible components, for the Hilbert scheme H(3,g) has roughly -g/3 irreducible components, where g is the arithmetic genus. This required classifying all the triple lines in P3, a classification which helped Dawei Chen to work out all the Mori theory of the moduli space of twisted cubic curves [Mori's program for the Kontsevich moduli space, Int. Math. Res. Not. (2009)]. Using concrete deformations, I showed that in fact there are certain curves which lie in the closure of all the components. This result was surprising at the time, for Martin-Deschamps and Perrin had recently proved that these Hilbert schemes are almost never reduced [Ann. Scient. \'Ec .Norm. Sup. 29 (1996)] and they expected disconnectedness as well.Subextremal curves [Manuscripta Math. 94 no. 3 (1997) 303-317.]
In the short and elegant paper [C. R. Acad. Sci. Paris 317(1993)], 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. Phillipe Ellia proved upper bounds on the Rao function for curves which are not extremal and coined the phrase ``subextremal'' for those curves achieving his bounds [Bolletino della U.M.I. 9-A (1995)]. Exploiting a new invariant (called the spectrum for space curves developed in the Ph.D. thesis of Enrico Schlesinger [The spectrum of projective curves, U.C. Berkeley PhD thesis, 1996. ], showed that his bounds could be improved slightly and classified all the curves achieving equality. I also showed that one can go no further, that there aren't tight bounds on curves which are not subextremal. Moreover, the methods of this paper work in finite characteristic, extending Martin-Deschamps and Perrin's original bound to this setting.Degrees of generators of ideals [Comm. Algebra 26 (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 (1998) 135-141.]
In [Complex manifold geography in dimension 2 and 3, J. Diff. Geom. 30 (1989)], 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, thanks to work of Persson [On Chern invariants of surfaces of general type, Compositio Math. 43 (1981)].In the present work, we prove a bound on 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. Chang and Kim have proved better bounds under the additional assumption that threefolds have principal Picard group generated by a very ample divisor [The Euler number of certain primitive Calabi-Yau threefolds, Math. Proc. Cambridge Philos. Soc. 128 (2000)].
Bounds on Multisecants [Collect. Math. 49 no. 2-3 (1998) 447-463.]
Here we give new insights into several known results about arithmetically Cohen-Macaulay (ACM) subvarieties V in Pn of codimension two. Using a short linking trick, we give a sharp upper bound on the largest degree of generators for the total ideal IV, recovering the main theorem of Campanella [J. Algebra 101 (1986)]. When V is integral, the proof yields a stronger bound along with information about the gamma character studied by Martin-Deschamps and Perrin [Sur la Classification des Courbes Gauches, Asterisque 184-185 (1990)].In the case of curves in P3, this theorem recovers as corollaries the results in papers of Sauer [Math. Ann. 272 (1985)], Gruson and Peskine [Lecture Notes in Mathematics 687 (1978)] and Maggioni and Ragusa [Invent. Math. 91 (1988)]. In fact, our examples of sharpness for the theorem improve on these papers in that (a) the method works in finite characteristic, (b) the curves lie on smooth surfaces of minimal degree and (c) the curves have maximal order multisecant lines. Our construction uses some local algebra which concretely describes the singular locus of a blow-up along a codimension two subvariety, recovering a result of O'Carroll and Valla [Comm. Algebra 25 (1997)]. Another consequence is that general arithmetically Cohen-Macaulay subscheme of codimension two has singular locus of codimension > 3, as shown earlier by Chang's filtered Bertini theorem [J. Reine Angew. Math. 397 (1989)].
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 [12] 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.]
The Lazarsfeld-Rao property for even liaison classes L of codimension two subvarieties V in Pn was interpretted as saying that this classes are partially ordered by domination and have a unique minimal element up to deformation in work of Martin-Deschamps and Perrin [Sur la Classification des Courbes Gauches, Asterisque 184-185, 1990]. Generally speaking, algebraic geometers are more interested in integral or smooth subschemes, as these have more geometry to them. 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 [Linkage of General curves of large degree, in Lecture Notes in Mathematics 997, Springer-Verlag (1983)].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 [Sur les bornes du module de Rao, A. R. Acad. Sci. Paris, 317, (1993); Le schema de Hilbert des courbes gauches localement Cohen-Macaulay n'est (presque) jamais reduit, Ann. scient. Éc. Norm. Sup. 29 (1996)] and the subextremal curves 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 by Rich Liebling [Classification of space curves using initial ideals, U.C. Berkeley PhD thesis, 1996.]) 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 [Stable reflexive sheaves II, Invent. Math. 66 (1982)]. 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. We give examples showing sharpness for some small values of the Chern classes.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 been shown in the Ph.D. thesis of Samir [Sur le schˇma de Hilbert des courbes gauches de degrˇ d et genre = half(d-3)(d-4), Univ. de Paris Sud Thesis, 1998.] 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.]
Fix a complete Riemannian metric g on Rn and a local diffeomorphism f:Rn -> Rn. We show that if the height function given by taking the inner product with f and v satisfies the Palais-Smale condition relative to g for each for each nonzero v in Rn, then f is bijective. Our method substantially improves the classic global inversion theorem of Hadamard. Our result was recently strengthened by E. Cabral Balreira [Foliations and global inversion, Comment. Math. Helv. 85 (2010) ]. 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 [Collectanea Math. 54 (2003) 327-340.]
We study families of curves which lie on the doubling of a smooth surface X inside a smooth 3-fold, following the method introduced by Hartshorne and Schlessinger [Curves in the double plane, Comm. Alg. 28 (2000)], 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. This description is useful in classifying non-reduced curves on surfaces of low degree.The Hilbert schemes of degree four curves [Compositio Math. 139 (2003) 169-196.]
This paper builds on and extends on my earlier paper [The Hilbert schemes of degree three curves, Ann. Sci. Ecole Norm. Sup. 30 (1997)] by classifying all locally Cohen-Macaulay space curves of degree four and arithemetic genus g: we describe all the irreducible components of the Hilbert schemes H(4,g) and the list is substantially longer than for degree 3 curves. 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 [Un contre-exemple sur les familles de courbes gauches, Comm. Alg. 28 (2000)].Holomorphic injectivity and the Hopf map [Geom. and Funct. Anal. 14 (2004) 1339-1351.]
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.Deformations of space curves: Connectedness in Hilbert schemes [Rend. Sem. Mat. Univ. Politec. Torino 64 (2006) 433-450.]
I was invited to give a talk in honor of Paolo Valebrega for his 60th birthday conference in February 2005 and gave a survey talk on the question as to whether the Hilbert schemes H(d,g) of locally Cohen-Macaulay curves of degree d and genus g in P3 are connected. Afterwards I wrote up the survey and used the opportunity to prove some new results which related my topic to the man being honored, Paolo Valabrega. Being a survey paper, I will describe the contents of the sections:1. Geography of the Hilbert scheme. Here I described explained work of Hartshorne, Martin-Deschamps and Perrin which describe the pairs (d,g) for which H(d,g) is non-empty.
2. Connectedness Results. This section describes known results of several authors regarding connectedness of flat families of space curves.
3. The Hilbert scheme H(4,-99). In an effort to convey the state of the art of this difficult problem, I described the contents of my paper with Enrico Schlesinger [Hilbert schemes of degree four curves, Compositio Math. 139 (2003)] and as an example described the Hilbert scheme H(4,-99) and its 529 irreducible components.
4. Curves on the double quadric. This section describes in detail a new technique for constructing certain families of curves from section 3, based on another paper with Enrico Schlesinger [Curves on a double surface, Collectanea Math. 54 (2003)].
5. Subcanonical curves. Here I classified the subcanonical curves appearing in H(4,-99). There are 18 such families, and one of these contains subcanonical curves that were discovered by Valabrega and Chiantini 20 years earlier [Subcanonical curves and complete intersections in projective 3-space, Ann. Mat. Pura Appl. 138 (4) (1984)].
On Kulikov's problem [Arch. Math. (Basel) 89 (2007) 385-389.]
There are many generalizations of the Jacobian Conjecture, one of which is the following question: must every etale morphism F: X -> Cn which is surjective modulo codimension 2 with X simply connected be birational? Kulikov used an old example of Zariski to show that the answer in general negative [Generalized and local Jacobian problems, Russian Acad. Sci. Izv. 41 (1993)]. Since F is dominant, there is a hypersurface D in Cn for which the restriction map F: X - F-1 (D) -> Cn - D is a d-sheet covering map. If X is a simply connected manifold andthe closure of D in Pn meets the hyperplane at infinity transversely
D has at worst normal crossing singularities away from a set of codimension 3
then either d=1 or d is infinite. The result is sharp in that both cases actually occur (d=1 is achieved by the identity map while d=infinity is achieved by the complex exponential map), but d=1 is the only possibility for algebraic maps. The proof uses methods of algebraic topology and FultonÕs solution to the Zariski conjecture [On the fundamental group of the complement of a node curve, Ann. Math. 111 (1980)].Birationality of etale maps via surgery [J. Reine Angew. Math. 627 (2009) 83-95.]
The Jacobian Conjecture states that a polynomial map F: Cn -> Cn with nowhere vanishing Jacobian is a polynomial automorphism. In this paper we continue our program of understanding birationality of etale mappings and the phenomenon of local versus global principle [On KulikovÕs problem, Arch. Math. (Basel) 89 (2007); Holomorphic injectivity and the Hopf map, Geom. And Funct. Anal. 14 (2004); Global inversion via the Palais-Smale condition, Discrete Contin. Dyn. Syst. 8 (2002)].An (n-1)-dimensional submanifold A in Rn nicely bounds a closed subset D in Rn if D is the boundary of A, each connected component of the closure of A contains exactly one connected component of D, and D is be the closure of an (n-2)-dimensional submanifold D0 in Rn with singular locus S = D - D0 of codimension at least 4 with the closure of A locally diffeomorphic to a half-space along D0. The first result is the following: Let D in Cn be a smooth connected non-bifurcated hypersurface and let X be a simply connected manifold. Suppose that F: X -> Cn is a local diffeomorphism, and the restriction F: X - F-1 (D) -> Cn - D is a d-sheet covering map. then d = 1 or d is infinite.
As a corollary we show that if F: Cn -> Cn is a polynomial map with nonvanishing Jacobian and the number of points in all fibers over the points outside D which is a non-bifurcated hypersurface in Cn are equal to the degree of F, then F is an automorphism. Over the reals we prove the following: let F:X -> Rn be a local diffeomorphism, X be a connected manifold with H_1(X,Z)=0, and D in Rn be a closed set for which the restriction F: X - F-1 (D) -> Rn - D is a d-sheeted covering. If D is nicely bounded by a connected (n-1)-dimensional submanifold A such that Rn - A is simply connected, then d can be 1, 2 or infinity. In fact, all possible values of d can be achieved (the case d=2 is achieved by an collared embedding of the Klein bottle in R4 ). The proof uses algebraic topology and surgery theory.