Research Summary |
---|
Much of my initial research in algebraic geometry was motivated by the problem of finding sharp vanishing theorems for the higher cohomology of stable rank two bundles on \(\mathbb P^3\). In attacking this from various directions, I've learned about linkage theory, vanishing theorems, cohomological bounding techniques and Hilbert schemes. Later I branched out to other topics, such as Macaulay's growth bound for polynomial rings (see paper with Cristina Blancafort) and the Jacobian conjecture (see papers with Frederico Xavier). More recently I've done some work on Hilbert schemes with Dawei Chen and Noether-Lefschetz theory with applications to finding local Picard groups (see papers with John Brevik). Papers are in reverse chronological order with reference numbers referring to the bibliography at the end. Along with the journal citation and brief description of the results, there is a link at the top to some electronic version of the paper if available (several go to the math arXiv). |
Publications |
Offprint Algebraic Geometry II [Book review], Bull. Amer. Math. Soc. (N.S.) 57 (2020), 133--143. David Mumford's "Algebraic Geometry I" was published in 1976, but he had written notes for two follow up books on schemes and moduli problems. His notes on scheme theory were about 2/3 done at the time, but lay dormant until his student Tadao Oda prepared them for publication, with further exercises and examples by Ching-Li Chai, Vikraman Balaji and D.S. Nagaraj. There is also an additional section on Seshadri constants by Nagaraj. My book review covers the historical context in which scheme theory arose and then gives a description of the book itself. |
Reprint Moishezon's theorem and degeneration (with John Brevik), J. Algebra 544 (2020), 463--482. Let \(L\) be a tensor product of two very ample line bundles on a smooth threefold \(X\). Under the hypothesis \(H^0 (X, K_X (L)) \neq 0 \) we show that the restriction map \( r:{\rm {Pic}} X \to {\rm {Pic}} Y \) is an isomorphism for very general \( Y \) in the linear system \( |L| \). For such \( L \) this recovers the Noether-Lefschetz theorem of Moishezon from 1966, where he extended the original topological and Hodge-theoretic arguments of Lefschetz. We give a degeneration proof which almost completely algebraic. |
arXiv:1606.01977 Geometric divisors in normal local domains (with John Brevik), J. Pure Appl. Algebra 222 (2018), 464--476. A geometric local ring is the local ring \( A = {\mathcal O}_{X,x} \) for some normal point \( x \in X \) on a complex algebraic variety. If \( R = \hat A \) is the completion, there is a natural injective homomorphism \( i: {\rm {Cl}} \hookrightarrow {\rm {Cl}} R \) which need not be surjective. Fixing \( R \), Srinivas has asked about all possible images as the local ring \( A \) varies [Srinivas02]. We define a geometric divisor of \( R \) to be any element of such an image and ask some basic questions: (a) Given any finitely generated group \( G \subset {\rm {Cl}} R \), is there a geometric local ring \(B\) with \( \hat B \cong R \) and \( G = {\rm {Cl}} B \)? (b) Given \( \alpha_1, \dots, \alpha_r \in {\rm {Cl}} R \), is there a geometric local ring \( B \) with \( \hat B \cong R \) and \( \alpha_i \in {\rm {Cl}} B \) for each \( i \)? (c) Is every \( \alpha \in {\rm {Cl}} R \) a geometric divisor? (d) Do geometric divisors form a subgroup of \( {\rm {Cl}} R \)?
We have easy implications \( (a) \Rightarrow (b) \Rightarrow (c) \Rightarrow (d) \)
and conjecture that if \( R \) is the completion of a normal geometric local complete intersection ring and \( G \subset {\rm {Cl}} R \) is any finitely generated subgroup, then there is a geometric local ring \( B \) with \( \hat B \cong R \) and \( G = {\rm {Cl}} B \subset {\rm {Cl}} R \). We make some progress toward this conjecture with the following result: |
arXiv:1601.05846 Grothendieck-Lefschetz theorem with base locus (with John Brevik), Israel J. Math. 212 (2016), 107--122. Ravindra and Srinivas proved that if \( X \) is a normal projective variety of dimension \( \geq 4 \) and \(f: X \to \mathbb P^n \) is a morphism defined by an ample line bundle \( {\mathcal O}_X (1) \), then the restriction map \( {\rm {Cl}} X \to {\rm {Cl}} Y \) is an isomorphism for Zariski-general \( Y \in |H^0({\mathcal O}_X (1))| \) [RS1]. Earlier we proved that if \( Z \subset \mathbb P^N \) is any closed subscheme of codimension \( \geq 2 \) lying on a normal hypersurface, then a general hypersurface \( H \) containing \( Z \) of high degree has class group \( {\rm {Cl}} H \) freely generated by \( {\mathcal O}_H (1) \) and the supports of the codimension two irreducible components of \( Z \) [BN1]. The main theorem of this paper is a common generalization of these two results: Theorem. Let \( X \subset \mathbb P^N \) be a normal projective variety of dimension \( \geq 4 \) and let \( Z \subset X \) be a closed subscheme of codimension \( \geq 2 \) with \( {\mathcal I}_Z (d-1) \) generated by global sections. Assume that the irreducible components \( Z_i \subset Z \) of codimension two are not contained in the the singular locus of \( X \) and have generic embedding dimension at most \( \dim X - 1 \). Then the general member \( Y \in |H^0({\mathcal I}_Z (d))| \) is normal and the restriction map \( {\rm {Cl}} X \to {\rm {Cl}} Y \) is split injective with cokernel generated by the supports of the \( Z_i \). As an application, we show that the completion of any normal complete intersection singularity of dimension at least three is the completion of a unique factorization domain. |
arXiv:1403.2423 Srinivas' problem for rational double points (with John Brevik), Michigan Math. J. 64 (2015), 155--168. In his survey of geometric methods in commutative algebra, Srinivas [Srinivas02] raised the following interesting questions: Suppose that \( B \) is the completion of a geometric domain \( A \), meaning that \( A \) is an integral domain of finite type over \( \mathbb C \). 1. What subgroups appear as the image of the map on divisor class groups \( {\rm {Cl}} A \hookrightarrow {\rm {Cl}} B \) taken over all geometric domains \( A \) whose completion is isomorphic to \( B \)? 2. Is it possible to obtain \( \langle \omega_B \rangle \subset {\rm {Cl}} B \) as such an image? While there has been quite a bit of progress on problem 2, especially for complete intersection rings, almost no progress has been made on problem 1. In this paper we prove that for completions of local rings of surface rational double point singularities, the answer is all subgroups. |
Preprint Developments in Noether-Lefschetz theory (with John Brevik), Contemporary Mathematics 608 (2014) 21--50. This is a survey paper covering a vast amount of work in Noether-Lefschetz theory. We compare the ideas of the proofs and statements of the results, including what it known about the Noether-Lefschetz locus. The outline follows: 1. The Classical Theorem. Noether's original idea and statement from 1882 and the proof of Lefschetz from the 1920s, Moishezon's variant for algebraic homology classes (1967) and the generalization to finite characteristic due to Deligne, Grothendieck and Katz (SGA 7), and the stronger Grothendieck-Lefschetz theorem for higher dimensions. 2. New ideas: the infinitesimal approach. After reviewing variations of Hodge structures and some results about the moduli of projective hypersurfaces, we present the infinitesimal Noether theorem of Carlson, Green, Griffiths and Harris, the explicit Noether theorem (via Koszul cohomology) of Green, the degeneration proof of Griffiths and Harris, Ein's generalization of Noether's theorem to dependency loci of sections of vector bundles, Joshi's variant for singular surfaces, recent results of Ravindra and Srinivas about the class groups with normal ambient spaces, and our own variation of Noether's theorem with base locus. 3. Components of the Noether-Lefschetz locus. The density of general components of Ciliberto, Harris, Miranda and Green, the work of Green and Voisin on components of small codimension and Otwinowska's asymptotic result, the work of Ciliberto and Lopez on the distribution of codimensions, and Voisin's counterexample to the conjecture that there are finitely many special components. 4. An application. A recent application of Noether-Lefschetz theory to a question of V. Srinivas about class groups of local rings. |
Reprint Picard groups of normal surfaces (with John Brevik), J. Singul. 4 (2012) 154--170. We study the fixed singularities imposed on members of a linear system of surfaces in \( \mathbb P^3_{\mathbb C} \) by its base locus \( Z \). For a 1-dimensional subscheme \( Z \subset \mathbb P^3 \) with finitely many points \( p_i \) of embedding dimension three and \( d \gg 0 \), we determine the nature of the singularities \( p_i \in S \) for general \( S \in |H^0 (\mathbb P^3, I_Z (d))| \) and give a method to compute the kernel of the restriction map \( {\rm {Cl}} S \to {\rm {Cl}} {\mathcal O}_{S,p_i}. \) One tool developed is an algorithm to identify the type of an \( {\bf {A_n}} \) singularity via its local equation. We illustrate the method for representative \( Z \) and use Noether-Lefschetz theory to compute \( {\rm {Pic}} S. \) |
arXiv:0911.2221 Detaching embedded points (with Dawei Chen), Algebra Number Theory 6 no. 4 (2012) 731--756. Suppose that \( X \subset Y \subset \mathbb P^N \) differ at finitely many points: when is \( Y \) a flat limit of \( X \) union isolated points? Our main result says that this is possible when \( X \) is a local complete intersection of codimension 2 and the multiplicities are at most 3. We show by example that no hypothesis can be weakened: the conclusion fails for (a) embedded points of multiplicity greater than 3, (b) local complete intersections \( X \) of codimension greater than 2 and (c) non-local complete intersections of codimension 2. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus and show the smoothness of the Hilbert component whose general member is a plane curve union a point in \( \mathbb P^3 \). We also give a rigorous proof that the Hilbert scheme of rational quartic curves in \( \mathbb P^3 \) has 4 evident components, whose general members consist of (a) smooth irreducible rational quartics, (b) disjoint unions of lines and plane cubics, (c) the disjoint union of an elliptic quartic and a point and (d) the disjoint union of a plane quartic and 3 points. |
arXiv:0909.5170 The Hilbert scheme of a pair of codimension two linear subspaces (with Dawei Chen and Izzet Coskun), Comm. Algebra 39 no. 8 (2011) 3021--3043. Consider the general union of two linear subspaces \( L, M \subset \mathbb P^n \) of codimension 2. For general \( L, M \), these intersect in a linear subspace of codimension 4. As these vary, they form an irreducible component \( H \) of the Hilbert scheme. In this paper we show that \( H \) only intersects one other component of the Hilbert scheme, and that while the full Hilbert scheme is singular along this intersection, the component \( H \) is a smooth variety when considered by itself. We compute the Picard group of \( H \), along with effective and ample cones. For each divisor in the ample cone we work out the corresponding Mori model, and in particular show that \( H \) is a Mori dream space. |
arXiv:0806.1243 Noether-Lefschetz theorem with base locus (with John Brevik), Int. Math. Res. Not. IMRN 2011 no. 6, 1220--1244. The main result in this paper is the following: Theorem. Let \( Z \subset \mathbb P^3 \) be a closed subscheme which lies properly on a normal surface. Then the general surface \( S \) of high degree containing \( Z \) is normal and its class group is freely generated by \( {\mathcal O}_S (1) \) and the supports of the curve components of \( Z \). As special cases, we recover the classical Noether-Lefschetz theorem by taking \( Z \) to be empty, concluding that the general surface \( S \subset \mathbb P^3 \) of degree \( d > 3 \) is smooth with Picard group isomorphic to \( \mathbb Z \), generated by the hyperplane class. When \( Z \) is a smooth connected curve we recover a theorem of Lopez which has found many applications [Lopez91]. When \( Z \) is zero dimensional and non-reduced, we improve on Joshi's result that the general singular surface \( S \) satisfies the conclusion of the classic theorem [Joshi95]. There are also several applications to computing Picard groups. |
arXiv:0705.0524 Birationality of etale maps via surgery (with Laurence Taylor and Frederico Xavier), J. Reine Angew. Math. 627 (2009) 83--95. The Jacobian Conjecture states that a polynomial map \( F: \mathbb C^n \to \mathbb C^n \) 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 Kulikov's 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 \( \subset \mathbb R^n \) nicely bounds a closed subset \( D \subset \mathbb R^n \) 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 \( D_0 \subset \mathbb R^n \) with singular locus \( S = D - D_0 \) of codimension at least 4 with the closure of \( A \) locally diffeomorphic to a half-space along \( D_0 \). The first result is the following: Theorem 1. Let \( D \subset \mathbb C^n \) be a smooth connected non-bifurcated hypersurface and let \( X \) be a simply connected manifold. Suppose that \( F: X \to \mathbb C^n \) is a local diffeomorphism, and the restriction \( F: X - F^{-1} (D) \to \mathbb C^n - D \) is a d-sheet covering map. then \( d = 1 \) or \( d = \infty \). As a corollary we show that if \( F: \mathbb C^n \to \mathbb C^n \) 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 \( \mathbb C^n \) are equal to the degree of \( F \), then F is an automorphism. Over the reals we prove the following: Theorem 2. let \( F : X \to \mathbb R^n \) be a local diffeomorphism, \( X \) a connected manifold with \( H_1(X,Z)=0 \), and \( D \subset \mathbb R^n \) a closed set for which the restriction \( F: X - F^{-1} (D) \to \mathbb R^n - D \) is a d-sheeted covering. If \( D \) is nicely bounded by a connected \( (n-1) \)-dimensional submanifold \( A \) such that \( \mathbb R^n - A \) is simply connected, then \( d = 1,2 \) or \( \infty \). In fact, each of these values of \( d \) can be achieved (the interesting case \( d=2 \) is achieved by an collared embedding of the Klein bottle in \( \mathbb R^4 \). The proof uses algebraic topology and surgery theory. |
Reprint On Kulikov's problem (with Frederico Xavier), Arch. Math. (Basel) 89 (2007) 385-389. There are many generalizations of the Jacobian Conjecture, including the following due to V. Kulikov [K93]: must every etale morphism \( F: X \to \mathbb C^n \) 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 [K93]. Since F is dominant, there is a hypersurface \( D \subset \mathbb C^n \) for which the restriction map \( F: X - F^{-1} (D) \to \mathbb C^n - D \) is a d-sheet covering map. In this short note we prove the following theorem. Theorem. If \( X \) is a simply connected manifold and (a) the closure of \( D \) in \( \mathbb P^n \) meets the hyperplane at infinity transversely (b) \( D \) has at worst normal crossing singularities away from a set of codimension 3 then either \( d=1 \) or \( d = \infty \).The result is sharp in that both cases actually occur: \( d=1 \) is achieved by the identity map while \( d=\infty \) is achieved by the complex exponential map. For algebraic maps, \( d=1 \) is the only possibility, giving an application to the Jacobian conjecture, though note that if the conjecture is true, then \( D \) may be taken empty. The proof uses methods of algebraic topology and Fulton's solution to the Zariski conjecture [Fulton80]. |
Preprint Deformations of space curves: Connectedness in Hilbert schemes, Rend. Sem. Mat. Univ. Politec. Torino 64 no. 4 (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 \( \mathbb P^3 \) 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 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, including Hartshorne's thesis [Hartshorne66] and Lawrence Ein's result about nonspecial curves [Ein86]. 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, one of which contains subcanonical curves that were discovered by Valabrega and Chiantini 20 years earlier [CV84]. |
arXiv:0501196 Holomorphic Injectivity and the Hopf map (with Frederico Xavier), Geom. and Funct. Anal. 14 no. 6 (2004) 1339--1351. In this paper we ask under what conditions an etale morphism \( F: X \to \mathbb C^n \) 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. Most of the proof is dedicated to showing that a natural construction of tangent vectors is continuous with an Arzela-Ascoli type argument. |
arXiv:0112167 The Hilbert Schemes of Degree Four Curves (with Enrico Schlesinger), Compositio Math. 139 no. 2 (2003) 169-196. In this paper we determine the irreducible components of the Hilbert schemes \( H_{4,g} \) of locally Cohen-Macaulay space curves of degree 4 and arithmetic genus \( g \) for all \( g \leq 6 \). There are roughly \( g^2 / 24 \) such components for varying dimensions, most of which correspond to families of multiplicity structures supported on lines. We give deformations which shows that these Hilbert schemes are always connected. For \( g \leq -3 \) we exhibit a component which is disjoint from the component of extremal curves, thereby giving a counterexample to a conjecture of Ait-Amrane and Perrin [AP00]. |
arXiv:0112168 Curves on a double surface (with Enrico Schlesinger), Collect. Math. 54 no. 3 (2003) 327-340. Let \( X \subset T \) be the double structure of smooth projective surface \( F \subset T \) in a smooth threefold. A locally Cohen-Macaulay curve \( C \subset X \) naturally gives rise to two effective divisors on \( F \), namely the largest divisor \( P \subset C \cap F \) and the residual curve \( R \) to \( C \cap F \) in \( C \). We show that under suitable hypotheses a general deformation of \( R \) and \( P \) on \( F \) lifts to a deformation of \( C \) on \( X \). We give several applications to the study of Hilbert schemes of locally Cohen-Macaulay space curves. This result is an important tool in our classification of degree four locally Cohen-Macaulay curves in \( \mathbb P^3 \). |
Preprint Global Inversion via the Palais-Smale Condition (with Frederico Xavier), Disc. Cont. Dyn. Syst. - Ser. A 8 no. 1 (2002) 17-28. Fix a complete Riemannian metric \( g \) on \( \mathbb R^n \) and a local diffeomorphism \( f: \mathbb R^n \to \mathbb R^n \). 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 \( 0 \neq v \in \mathbb R^n \), then \( f \) is bijective. Our method substantially improves the classic global inversion theorem of Hadamard, extended to Banach spaces by Plastock [P74]. One interesting application is the following. For a smooth map \( f \) and setting \( S_v = \{ Df(x)^* v: x \in \mathbb R^n\} \), it is easy to see that \( f \) is locally invertible if and only if \( 0 \not \in S_v \) for each \( 0 \neq v \in \mathbb R^n \). Our result says that if \( 0 \not \in {\overline S_v} \) for each \( 0 \neq v \in \mathbb R^n \), then \( f \) is globally invertible. Our result was recently strengthened by E. Cabral Balreira [B10]. |
Preprint A remark on connectedness in Hilbert schemes, Comm. Algebra 28 no. 12, (2000) 5745-5747. (Special issue in honor of Robin Hartshorne) In this short note we show that for any pair \((d,g)\) for which the Hilbert scheme \(H(d,g)\) has subextremal curves, the subextemal curves can be deformed to extremal curves. As a corollary, we find that the Hilbert scheme \(H(d,g)\) is connected when \(g=1/2 (d-3)(d-4) + 1\). The same result is obtained in the thesis of Samir Ait-Amrane using the heavy machinery of variation of module structures [HMDP99] and a computer calculation [A98]. Samir's thesis also proves connectedness when \(g=1/2 (d-3)(d-4)\), a more difficult case. |
On the Second Section of a Rank Two Bundle (with Margherita Roggero and Paolo Valabrega),
Comm. Algebra 28 no. 12, (2000) 5531-5540. (Special issue in honor of Robin Hartshorne) For stable rank two reflexive sheaves \({\mathcal E}\) on \(\mathbb P^3\), Hartshorne gave a sharp upper bound on the least number \(a\) such that \({\mathcal E}(a)\) must have a nonzero section in terms of the Chern classes of \( {\mathcal E} \) [Hartshorne82]. In the present paper we give a similar bound (in terms of the Chern classes of \({\mathcal E}\) and \(a\) for the least number \(b\) such that \({\mathcal E}(b)\) has two independent sections. We give examples showing sharpness for some small values of the Chern classes. |
Reprint Bounds on the Rao Function (with Rosa Maria Miró-Roig), J. Pure Appl. Algebra 152 no. 1-3 (2000) 253-266. (Special issue in honor of David Buchsbaum) Similar to "Subextremal Curves" below, the aim of this paper is to bound the Rao function \(r_C (n) = H^1({\mathcal I}_C (n))\) associated to a curve \(C \subset \mathbb P^3\). 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 [MDP93,MDP96] 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 \(C\) developed by Rich Liebling in his thesis [Liebling96] called the triangle diagram, which arises naturally from the generic initial ideal. The bounds we obtain are sharp for \(s \leq 5\). |
arXiv:9504008 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 \(\mathbb P^n\) 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 [MDP90]. 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 [LR83]. In one application, Paxia and Ragusa use this result (the version from my Ph.D. thesis) to characterize the irreducible smooth Buchsbaum curves in \( \mathbb P^3 \) [PR95]. |
Reprint Hilbert Polynomials over Artinian Rings (with Cristina Blancafort), Illinois J. Math. 43 no. 2 (1999) 338-349. Given an Artinian ring R and a homogeneous ideal \(I \subset S = R[x_0,x_1,\dots,x_n]\), consider the associated Hilbert function \(H(k) = \dim_k (S/I)_k\). In this paper we describe all such functions \(H\) arising in this way, extending the original result of Macaulay [Macaulay27] in the case when \(R\) is a field and as a corollary we characterize the associated Hilbert polynomials. We also extend Gotzmann's regularity and persistence theorems [Gotzmann78] to this setting. The inspiration for this work was Green's geometric treatment for these topics [Green89]. |
Bounds on Multisecants,
Collect. Math. 49 no. 2-3 (1998) 447-463. (Volume in memory of Fernando Serrano)\( {}^\dagger \) Here we give new insights into several known results about arithmetically Cohen-Macaulay (ACM) subvarieties V in \(\mathbb P^n\) of codimension two. Using a short liaison trick, we give a sharp upper bound on the largest degree of generators for the total ideal \(I_V\), recovering a theorem of Campanella [Campanella86]. When \(V\) is integral, the proof yields a stronger bound along with information about the gamma character studied by Martin-Deschamps and Perrin [MDP90]. In the case of curves in \(\mathbb P^3\), this theorem recovers as corollaries the results in papers of Sauer [Sauer85], Gruson and Peskine [GP78] and Maggioni and Ragusa [MR88]. In fact, our examples of sharpness for the theorem improve on these papers in that (a) the method works in arbitrary 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 [OV97]. 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 [Chang89]. \( \dagger \) It was from Fernando (Ferran) Serrano that I first learned algebraic geometry at U.C. Berkeley while he was a postdoc there in 1988. |
Reprint Bounds on \(c_3\) for Threefolds (with Mei-Chu Chang and Hoil Kim), Manuscripta Math. 97 no. 2 (1998) 135-141. In [Hunt89], Hunt initiated a study of the geography of threefolds, the study of which Chern triples \((c_1,c_2,c_3)\) 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 [Persson81]. In this 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 improve these bounds under the additional assumption that threefolds have principal Picard group generated by a very ample divisor [CK00]. |
Degrees of Generators of Ideals,
with Heath Martin and Juan Migliore, 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. |
Reprint Subextremal curves, Manuscripta Math. 94 no. 3 (1997) 303-317. In the short elegant paper, Martin-Deschamps and Perrin gave a sharp upper bound on the Rao function \(r_C (n) = h^1({\mathcal I}_C (n))\) of a space curve \(C\) in terms of the degree \(d\) and the arithmetic genus \(g\) of \(C\) [MDP93]. 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 [Ellia95]. Exploiting a new invariant, the spectrum for space curves due to Enrico Schlesinger [Schlesinger1,Schlesinger2], I improved his bounds (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. |
arXiv:9603011 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 [Hartshorne66], however connectedness for the open subset corresponding to 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 [Migliore86]. 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 \(\mathbb P^3\), a classification which was used by Chen in working out the Mori theory of the moduli space of twisted cubic curves [Chen09]. Using concrete deformations, I showed that in fact there are certain curves which lie in the closure of all the components, and hence that these Hilbert schemes are connected. This result was surprising at the time, for Martin-Deschamps and Perrin had recently proved that these Hilbert schemes are almost never reduced [MDP96] and they expected disconnectedness as well. |
Reprint Smooth Curves Linked to Thick Curves, Compositio Math. 106 no. 2 (1997) 203-210. A conjecture of Martin-Deschamps and Perrin [MDP92] 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 infinitesimal neighborhood \(W^d\) (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. |
arXiv:9412013 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 [Rao1,Rao2] to a bijection \[ \left\{\begin{matrix} \mbox{Even liaison classes of} \\ \mbox{codimension two subschemes} \\ V \subset \mathbb P^n \end{matrix}\right\} \cong \left\{\begin{matrix} \mbox{Stable equivalence classes of } \\ \mbox{reflexive sheaves } {\mathcal E} \mbox{ on } \mathbb P^n \mbox{ with} \\ H^1_*({\mathcal E})=0 \mbox{ and } {\mathcal {Ext}}^1({\mathcal E},{\mathcal O})=0 \end{matrix}\right\}. \] We also show that the structure theorem of Ballico, Bolondi and Migliore [BBM91] carries over and even the algorithm of Martin-Deschamps and Perrin for locating the minimal subscheme in an even liaison class [MDP90]. |
Bibliography | |
---|---|
[A98] | S. Ait-Amrane, Sur le schema de Hilbert des courbes gauches de degre \( d \) et genre \( g = \frac{1}{2} (d-3)(d-4) \), Thesis, Univ. de Paris Sud, Centre d'Orsay, 1998. |
[AP00] | S. Ait-Amrane and D. Perrin, Un countreexemple sur les familles de courbes gauches, Comm. Algebra 28 (2000) 6003-6015. |
[B10] | E. Cabral Balreira, Foliations and global inversion, Comment. Math. Helv. 85 (2010) 73-93. |
[BBM91] | E. Ballico, G. Bolondi and J. Migliore, The Lazarsfeld-Rao problem for liaison classes of two-codimensional subschemes of \(\mathbb P^n\), Amer. J. Math. 113 (1991) 117-128. |
[BN1] | J. Brevik and S. Nollet, Noether-Lefschetz theorem with base locus, Int. Math. Res. Not. 2011 (2011) 1220--1244. |
[Campanella86] | G. Campanella, Standard bases of perfect homogeneous polynomial ideals of height 2, J. Algebra 101 (1986) 47-60. |
[Chang89] | M. C. Chang, A filtered Bertini-type theorem, J. Reine Angew. Math. 397 (1989) 214-219. |
[CK00] | M. C. Chang and H. Kim, The Euler number of certain primitive Calabi-Yau threefolds, Math. Proc. Cambridge Philos. Soc. 128 (2000) 79-86. |
[Chen09] | D. Chen, Mori's program for the Kontsevich moduli space \({\overline {\mathcal M}}_{0,0} (\mathbb P^3,3)\), Int. Math. Res. Not. IMRN 2008, Art. ID rnn 067, 17pp. |
[CV84] | L. Chiantini and P. Valabrega, Subcanonical curves and complete intersections in projective 3-space, Ann. Mat. Pura Appl. 138 (4) (1984). |
[Ellia95] | P. Ellia, On the cohomology of projective space curves, Bolletino della U.M.I. 9-A (1995) 593-607. |
[Fulton80] | W. Fulton, On the fundamental group of the complement of a node curve, Ann. of Math. (2) 111 (1980) 407-409. |
[Gotzmann78] | G. Gotzmann, Eine Bedingung fur die Flachheit und sas Hilbertpolynom eines graduierten Ringes, Math. Z. 158 (1978) 61-70. |
[Green89] | M. Green, Restriction of linear series to hyperplanes, and some results of Macaulay and Gotzmann, Algebraic curves and projective geometry proceedings (Trento 1988) 76-86, Lecture Notes in Math. 1389, Springer, Berlin, 1989. |
[GP78] | L. Gruson and C. Peskine, Genre des courbes de l'espace projectif, Algebraic Geometry (Proc. Sympos. Univ. Tromso, Tromso 1977) 31-59, Lecture Notes in Math. 687, Springer, Berlin, 1978. |
[Hartshorne66] | R. Hartshorne, Connectedness of the Hilbert scheme, Inst. Hautes Études Sci. Publ. Math. No. 29 (1966) 5-48. |
[Hartshorne82] | R. Hartshorne, Stable reflexive sheaves II, Invent. Math. 66 (1982) 165-190. |
[HMPD99] | R. Hartshorne, M. Martin-Deschamps and D. Perrin, Triades et familles de courbes gauches, Math. Ann. 315 (1999) 397-468. |
[Hunt89] | B. Hunt, Complex manifold geography in dimension 2 and 3, J. Diff. Geom. 30 (1989) 51-153. |
[Joshi95] | K. Joshi, A Noether-Lefschetz theorem and applications, J. Alg. Geom. 4 (1995) 105-135. |
[K93] | V. Kulikov, Generalized and local Jacobian problems, Russian Acad. Sci. Izv. 41 (1993). |
[LR83] | R. Lazarsfeld and A. P. Rao, Linkage of general curves of large degree, Algebraic geometry-open problems (Ravello, 1982) 267-289, Lecture Notes in Math. 997, Springer, Berlin, 1983. |
[Liebling96] | R. Liebling, Classification of space curves using initial ideals, Thesis (Ph.D.)-University of California, Berkeley. 1996. 131 pp. |
[Lopez91] | A. Lopez, Noether-Lefschetz theory and Picard groups of projective surfaces, Memoirs of the American Mathematical Society 438 (1991). |
[Macaulay27] | F. S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927) 531-555. |
[MR88] | R. Maggioni and A. Ragusa, The Hilbert function of generic plane sections of curves in \(\mathbb P^3\), Invent. Math. 91 (1988) 253-258. |
[MDP90] | M. Martin-Deschamps and D. Perrin, Sur la Classification des Courbes Gauches, Astérisque 184-185 (1990). |
[MDP92] | M. Martin-Deschamps and D. Perrin, Construction de Courbes Lisses: un Théoème de Bertini, Laboratoire de Mathematiques de l'École Normale Supereure 22 (1992). |
[MDP93] | M. Martin-Deschamps and D. Perrin, Sur les bornes du module de Rao, C. R. Acad. Sci. Paris 317 (1993) 1159-1162. |
[MDP96] | M. Martin-Deschamps and D. Perrin, Le Schéma de Hilbert des courbes gauches localement Cohen-Macaulay n'est (presque) jamais réduit, Ann. Scient. Éc. Norm. Sup. 29 (1996) 757-785. |
[Migliore86] | J. Migliore, On linking double lines, Trans. Amer. Math. Soc. 294 (1986) 177-185. |
[OV97] | L. O'Carroll and G. Valla, On the smoothness of blow-ups, Comm. Algebra 25 (1997) 1861-1872. |
[PR96] | G. Paxia and A. Ragusa, Irreducible Buchsbaum curves, Comm. Algebra 23 (1995) 3025-3031. |
[Persson81] | U. Persson, Chern invariants of surfaces of general type, Compositio Math. 43 (1981) 3-58. |
[P74] | R. Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc. 200 (1974) 169-183. |
[Rao79] | A. P. Rao, Liaison among Curves in \(\mathbb P^3\), Invent. Math. 50 (1979) 205-217. |
[Rao81] | A. P. Rao, Liaison equivalence classes, Math. Ann. 258 (1981) 169-173. |
[RS1] | G. V. Ravindra and V. Srinivas, The Grothendieck-Lefschetz theorem for normal projective varieties, J. Algebraic. Geom. 15 (2006) 563--590. |
[Sauer85] | T. Sauer, Smoothing projectively Cohen-Macaulay space curves, Math. Ann. 272 (1985) 83-90. |
[Schlesinger1] | E. Schlesinger, The spectrum of projective curves, Thesis (Ph.D.)-University of California, Berkeley. 1996. 147 pp. |
[Schlesinger2] | E. Schlesinger, On the spectrum of certain subschemes of \(\mathbb P^N\), J. Pure Appl. Algebra 136 (1999) 267-283. |
[Srinivas02] | V. Srinivas, Some geometric methods in commutative algebra in Computational commutative algebra and combinatorics (Osaka, 1999), 231-276 , Adv. Stud. Pure Math. 33, Math. Soc. Japan, Tokyo, 2002. |