in
Commutative Algebra and its connections to Geometry
Olinda, Brazil
August 3 - August 14, 2009
Program and Schedule
(the two-week PASI program will take place at the Hotel 7 Colinas)
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Higher Fano manifolds
Carolina Araujo, IMPA (Brazil)
Abstract: Fano manifolds are smooth complex projective varieties having ample anti-canonical class. They form a very special class of varieties. For every fixed dimension, there is only a finite number of deformation types of Fano manifolds. So in principle they can be classified, and in fact classification has been achieved for dimension at most three. In a landmark paper, Mori showed that Fano manifolds contain rational curves through every point. Since then, many other special properties of Fano manifolds have been established, such as rationally connectedness.
A few years ago, de Jong and Starr introduced a special class of Fano manifolds, namely, Fano manifolds X with positive second Chern character ch2(X) > 0. They proved that, under some conditions, such X contains a rational surface through a general point. Unfortunately, very few examples of such higher Fano manifolds were known.
In this talk I will present a recent joint work with Ana-Maria Castravet on higher Fano manifolds. Our approach is via the study of rational curves of minimal degree through a general point x\in X. Our results in particular recover de Jong and Starr's theorem and enables us to find new examples of higher Fano manifolds.
Reflexivity and rigidity for complexes
Luchezar Avramov, U. Nebraska (USA)
Abstract: A notion of rigidity with respect to an arbitrary semidualizing complex $C$ over a commutative noetherian ring $R$ will be described. One of the main results is a characterization of homologically finite $C$-rigid complexes. Specialized to the case when $C$ is the relative dualizing complex of a homomorphism of rings of finite Gorenstein dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang concerning rigid dualizing complexes. Along the way, a number of new results concerning derived reflexivity with respect to $C$ are established. Noteworthy is the statement that derived $C$-reflexivity is a local property; it implies that a finite $R$-module $M$ has finite G-dimension over $R$ if $M_{\mathfrak m}$ has finite G-dimension over $R_{\mathfrak m}$ for each maximal ideal $\mathfrak m$ of $R$.
The talk is based on joint work with Srikanth Iyengar and Joseph Lipman.
Implicit equations of multigraded hypersurfaces
Nicolás Botbol, Universidad de Buenos Aires (Argentina)
Abstract: click here.
Normaliz: algorithms for rational cones and affine monoids
Winfried Bruns, U. Osnabrück (Germany)
Abstract: Normaliz is a computer program for the computation of Hilbert bases of rational cones and related tasks. Its main application in commutative algebra is the (equivalent) computation of integral closures of an affine monoid (or its algebra) in a surrounding lattice (or the corresponding group algebra). The first version was developed by the speaker and R. Koch in 1997-2001, and the new version 2 was implemented by B. Ichim. It has been improved in several aspects.
There are Normaliz packages for Singular and Macaulay 2. We will explain the algorithms and demonstrate the program and the packages.
This talk is a tribute to Wolmer Vasconcelos and his everlasting interest in the computation of integral closures.
Cremona geometry of plane curves
Ciro Ciliberto, U. Rome (Italy)
Tropicalisation of rational varieties
Alicia Dickenstein, Universidad de Buenos Aires (Argentina)
Abstract: The tropicalisation tau(V) of an affine or projective algebrai variety V is a polyhedral complex, together with intersection theoretic data, that encodes important information about V as its dimension, its degree, and its asymptotic directions. When V is a hypersurface, tau(V) carries the same information as the Newton polytope of a defining equation for V.
The problem of describing tau(V) for a rational variety in terms of a given rational parametrization has been recently studied by Sturmfels-Tevelev-Yu and Esterov-Khovanskii. Based on Kapranov's theorem, we present a naive approach to the problem using curve valuations, which allows us to slightly generalize their results. The advantage of this point of view is that even if we get complete results only when the polynomials defining the parametrization are generic with respect to their Newton polytopes, the proofs can be extended to deal with more general cases. This is joint work in progress with Bernard Mourrain.
The talk will be elementary.
Boij-Soederberg theory and the size of free resolutions
Daniel Erman, U. California at Berkeley (USA)
Abstract: The Buchsbaum-Eisenbud-Horrocks rank conjecture roughly says that the Koszul complex is the "smallest possible" free resolution of a graded module. Although Boij-Soederberg theory is based on the principle of only considering Betti diagrams up to scalar multiple, we will explain how the structure of Boij-Soederberg theory is sufficiently strong to prove new cases of the Buchsbaum-Eisenbud-Horrocks rank conjecture.
The Poincaré problem for subschemes invariant under Pfaff fields on projective spaces
Eduardo Esteves, IMPA (Brazil)
Abstract: The Poincaré problem, originating in a question raised by Poincari in a 1891 article, calls for bounds on the degrees of algebraic curves invariant under planar vector fields. We will review the history of the problem, the several attempts to give a solution to it and its extensions to higher-dimensional spaces. We will also present a recent development, an approach by the speaker and Steven Kleiman, using the Castelnuovo-Mumford regularity. Finally, we will present a new unpublished work, by the speaker and Joana Cruz, which generalizes the aforementioned work to higher-dimensional spaces.
Danilov Gizatullin surfaces and Ga-actions
Hubert Flenner, U. Bochum (Germany)
Abstract: (joint work with S.Kaliman (Miami) and M.Zaidenberg (Grenoble)) An affine surface (over C) is called a Danilov Gizatullin surface, if it can be written as a difference ∑n \ C, where ∑n is the Hirzebruch surface and C ⊆ ∑n is a section. We show how to classify on such a surface all conjugacy classes of A1-fibrations over an affine base. Such fibrations are essentially in 1-1 correspondence to Ga-actions on X. Our result shows in particular that the set of such conjugacy classes can depend on an arbitrary high number of parameters.
We also discuss the problem of classification of A1-fibrations on arbitrary affine surfaces. Such a surface is called Gizatullin, if if it admits a completion by a zigzag that is, by a linear chain of smooth rational curves. For non Gizatullin surfaces an A1-fibration over an affine base is unique if it exists. This holds also true for a large class of Gizatullin surfaces. In contrast, for the so-called special surfaces the conjugacy classes of A1-fibration can depend like in the case of Danilov-Gizatullin surfaces on an arbitrary high number of parameters. For general Gizatullin surfaces the problem of classification of such fibrations is still open.
On certain maximal curves over finite fields
Arnaldo Garcia, IMPA (Brazil)
Abstract: Over a finite field K with q2 elements, the Hermitian curve can be given by Yq+1 = Xq + X. The Hermitian curve has the biggest genus q(q-1)/2 possible for maximal curves over K, and moreover it is the unique maximal curve of that genus. By a result due to J.-P. Serre we know that if a curve C is covered by a maximal curve, then C is also a maximal curve. A kind of reciprocal of the result of Serre is the following interesting question: Is any maximal curve over K covered by the Hermitian curve? This question has a negative answer (due to Giulietti- Korchmaros) and we are going to present some maximal curves over K related to their construction of maximal curves that are not covered by the Hermitian.
Gotzmann coefficients of Hilbert functions
Anthony Geramita, Queen's U. (Canada)
On the hyperhomology of the small Gobelin for codimension 2
Xavier Gomez Mont, CIMAT (Mexico)
Abstract: The Gobelin and the small Gobelin are quasi-isomorphic complexes constructed from a commutative square of matrices over a ring. Using the small gobelin we exhibit patterns between its homology groups. This is joint work with Luis Nunez-Betancourt.
Cohen-Macaulayness versus vanishing of the first Hilbert coefficient of parameters:
towards a problem of Wolmer Vasconcelos
Shiro Goto, Meiji U. (Japan)
Abstract: click here.
Gonality of ACM curves in P3
Robin Hartshorne, U. California at Berkeley (USA)
Abstract: The gonality of an (abstract) curve is the least degree of a finite morphism to the projective line. We show that for (almost all) ACM curves in P3 the gonality is related to the projective embedding in the sense that there is a multisecant such that the projection from that line gives a map of least degree to P1. This is joint work with E. Schlesinger.
A Tight Closure Theory that Commutes with Localization in Equal Characteristic
Melvin Hochster, U. Michigan (USA)
Abstract: The talk will discuss joint work with Neil Epstein concerning a variant definition of tight closure both in positive characteristic and in equal characteristic 0, under the mild assumption that the ring is locally excellent. This new notion commutes with arbitrary localization! The characteristic p definition agrees with the original notion for systems of parameters and does not change the parameter test ideal. In all characteristics the new notion is smaller than the original: it is certainly strictly smaller in some cases in positive characteristic. The characteristic p notion contains the plus closure, and in all characteristics the new notion agrees with the original in graded cases in finitely generated graded algebras over a prime field. The new notion still captures colons and gives a theory of phantom homology similar to ordinary tight closure. All ideals are tightly closed over regular rings, and one has a Briancon-Skoda theorem. As is the case for ordinary tight closure, the new notion is persistent under arbitrary base change, and one can test the new notion modulo every minimal prime or by localizing and completing at all maximal ideals. The new notion gives rise to a new class of rings with the property that every ideal is tightly closed: this class agrees with the weakly F-regular rings in the Gorenstein case in characteristic p, contains all weakly F-regular rings, but is closed under localization. This work raises a host of new open questions.
A property of the ring of polynomials over a perfect field of characteristic p>0
Gennady Lyubeznik, U. Minnessota (USA)
Abstract: We will describe an adjointness property between the Frobenius pulback and pushforward and deduce some interesting consequences for local cohomology modules.
Blowup algebras and elimination theory
Claudia Polini, U. Notre Dame (USA)
Abstract: I will report on joint work with A. Kustin and B. Ulrich. We use elimination theory to find the defining equations of the Rees algebras of certain classes of ideals.
Oil fields and Hilbert schemes
Lorenzo Robbiano, U. Genova (Italy)
Abstract: New techniques for dealing with problems of numerical stability in computations involving multivariate polynomials allow a new approach to real world problems. Using a modeling problem for oil field production optimization as a motivation, we present several recent developments involving border bases of polynomial ideals. After recalling the foundations of border basis theory in the exact case, we present techniques for computing approximate border bases and stable order ideals. To get a deeper understanding for the algebra underlying this approximate world, we present recent advances concerning border basis and Groebner basis schemes which are subschemes of Hilbert schemes of zero-dimensional ideals.
Fontaine rings and local cohomology
Paul Roberts, U. Utah (USA)
Abstract: If R is a ring of mixed characteristic p, its Fontaine ring is a ring of positive characteristic defined from R from which R can be reconstructed (under certain conditions) up to p-adic completion. Since many theorems can be proven for rings of positive characteristic using the Frobenius map, it seems reasonable to attempt to do this for rings of mixed characteristic using this construction. In this talk we discuss this program, define several rings that come up in the process, and the questions that arise when using this method.
Counting compatibly Frobenius split ideals
Karl Schwede, U. Michigan (USA)
Abstract: I will discuss Frobenius split rings, with a focus on finiteness of compatibly split ideals (joint work with Kevin Tucker). Links with symbolic powers of canonical modules, F-singularities of pairs and log canonical centers will also be discussed.
Stanley decompositions of squarefree monomial ideals
YiHuang Shen, Purdue U. (USA)
Abstract: For a finitely generated Zn-graded module M over the polynomial ring k[x1,...,xn], one can consider its Stanley decompositions and Stanley depth sdepth(M). Stanley conjectured that sdepth(M) ≥ depth(M). The conjecture has been confirmed in several cases, but remains open in general. One obstacle to verifying this conjecture lies in the difficulty of computing the Stanley depth. Using a recent method due to Herzog, Vladoiu and Zheng, we determine the Stanley depth for several classes of squarefree monomial ideals.
Local cohomology with determinantal support
Anurag Singh, U. Utah (USA)
Abstract: We will describe some explicit computations of local cohomology modules, and discuss their implications to vanishing results. This is based on work in progres with Uli Walther.
Growth of Bass numbers
Janet Striuli, Fairfield U. (USA)
Abstract: Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k-vector space dimension of the cohomology module Exti(k, R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. In the talk I will discuss some experiments and preliminary results that are aimed at determining the growth patterns of these sequences.
Polynomial vector fields with algebraic trajectories
Israel Vainsencher, UFMG (Brazil)
Abstract: click here.
On the ideal theory of graphs (fifteen years later)
Rafael H. Villarreal, Instituto Politecnico Nacional/CINVESTAV (Mexico)
Abstract: We will introduce some of the results of [A. Simis, W. Vasconcelos, ---, On the ideal theory of graphs, J. Algebra 167 (1994), 389-416] and present some recent work inspired by this paper. We will consider algebras and ideals defined by a finite set of monomials of a polynomial ring over an arbitrary field. The emphasis will be on algebraic properties (such as the normality, torsion freeness, or Cohen-Macaulay property) of Rees algebras and monomial subrings defined by square-free monomials.
The a invariants of normal graded Gorenstein rings and varieties with even canonical class
Kei-ichi Watanabe, Nihon U. (Japan)
Abstract: Let $R=\oplus_{n\ge 0} R_n$ be a Noetherian normal graded ring with $R_0=k$ field and $\Proj(R)=X$. We always assume that GCD of $\{n\;|\; R_n\ne 0\}$ is $1$ but {\it not} assume $R$ is generated by elements of degree $1$.
Our question is as follows; Given a normal projective variety $X$, is there a (quasi) Gorenstein normal graded ring $R$ with $\Proj(R)=X$ ? If there exists one, then what is the possibility of $a(R)$ for fixed $X$ ?
Also, I will show that if there exists a (quasi) Gorenstein ring $R$ with $\Proj(R)=X$ and EVEN $a(R)$, then $X$ should have EVEN canonical class, that is, the class of $K_X$ is equal to that of $2D$ for some divisor $D$.
We also discuss about the possibility of the set $\calA (X)=\{a(R)\;|\; R$ is quasi Gorenstein, $\Proj(R)=X\}$.