Commutative Algebra and Algebraic Geometry Conference

View of Urbana

University of Illinois at Urbana-Champaign
Urbana, Illinois
November 4 - 6, 2011

Program and Schedule

All talks will be held in room 314 Altgeld Hall (Mathematics building).
Registration will take place on Friday from 3pm to 4pm in room 321 Altgeld Hall.
Coffee/tea breaks and light breakfast will also be served in room 321 Altgeld Hall.

Note: Time will fall back to standard time again on Sunday, November 6, 2011, when daylight saving time ends.

November 4
November 5
November 6
08:00-09:00   Light Breakfast
Light Breakfast
09:00-09:50   Hochster
10:00-10:50 Purnaprajna
11:00-11:50 Dao
Lunch Break

02:00-02:50 Szpiro
03:00-04:00 Coffee and Tea
Coffee and Tea
04:00-04:50 Ulrich
05:00-05:50 Li

Abstracts of Talks
  1. Picard Maximal Varieties
    Donu Arapura, Purdue University
    Abstract: Let me say that a smooth complex projective variety is Picard maximal if the space of algebraic cycles in cohomology is as large as possible, i.e. if it generates the $(p,p)$ part of the Hodge structure for each $p$. Part of the motivation for considering these is that various conjectures (Hodge, Grothendieck, Tate) hold automatically for such varieties, but I also think that their study has intrinsic value. I will briefly describe what is known in dimension 2, thanks to the efforts of Shioda and others. Then I will look at higher dimensional examples. The main result is that the Hilbert scheme of points on a Picard maximal surface is again Picard maximal.
  2. Asymptotic Growth of Saturated Powers of Ideals and Epsilon Multiplicity
    Dale Cutkosky, University of Missouri
    Abstract: We study the growth of saturated powers of ideals and modules. There are examples showing that the algebra of saturated powers of an ideal I in a noetherian local ring R is not a finitely generated R-algebra; As such, it cannot be expected that the ``Hilbert function'', giving the length of the R-module (I^k)^{sat}/I^k, is very well behaved for large k. However, it can be shown that it is bounded above by a polynomial in k of degree d, where d is the dimension of R. We show that for quite general domains, there is a reasonable asymptotic behavior of this length. We extend this to the case of modules to show that the epsilon multiplicity, defined by Ulrich, Validashti and Kleiman, exists as a limit over very general domains.
  3. Some homological conjectures revisited
    Hailong Dao, University of Kansas
    Abstract: The famous homological conjectures were formulated in the 50s and 60s and have had significant impacts on commutative algebra. Most of them can be phrased as surprising statements about finitely generated modules with finite projective dimension.
    In this talk we will survey some new statements that sound quite similar to the classical conjectures, but have distinctly different features and motivations. One notable difference is the switch from finite projective dimension property to the property that the classes of certain modules are 0 in the rational Grothendieck/Chow groups. We will discuss recently discovered connections to algebraic geometry and noncommutative geometry.
  4. A finiteness condition on local cohomology in positive characteristic Florian Enescu, Georgia State University
    Abstract: For a local ring R of positive prime characteristic, the local cohomology modules of the ring with support in the maximal ideal inherit a natural Frobenius action. The talk with discuss the lattice of submodules of the local cohomology that are compatible with the Frobenius action. It is known that this lattice contains remarkable information about the structure of the ring R. A conjecture regarding it will be formulated, and recent progress on it will be presented.
  5. Arithmetically Cohen-Macaulay bundles on hypersurfaces
    Mohan Kumar, Washington University
    Abstract: The talk is based on some joint work with A. P. Rao and G. V. Ravindra. The relevant preprints are available at arXiv:math.AG/0507161 , arXiv:math/0611620 and arXiv:math/1005.3990. The first appeared in Commentari Math. Helv., the second in IMRN and the third in Fields Inst. Comm. We will also touch on some work in progress.
    A vector bundle on a polarized projective variety (X;L) is called Arithmetically Cohen-Macaulay if all its middle cohomologies in all twists by powers of L vanish. A famous criterion of G. Horrocks states that a vector bundle on projective space is a direct sum of line bundles if and only if it is arithmetically Cohen-Macaulay (with respect to the usual polarization). It is well known that this criterion fails for other varieties, in particular for hypersurfaces in projective spaces. In my talk I will discuss the following results proved in the above articles. Any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least three in P5 or on a general hypersurface of degree at least six in P4 must be split.
    It is also known that for general quintic threefolds, rank two ACM bundles are rigid. So it leads to an interesting enumerative problem of counting these (upto twists). I will discuss what is known and what is still to be done.
  6. Diagonal F-threshold, Hilbert-Kunz multiplicity and socle degrees of Frobenius powers
    Jinjia Li, University of Louisville
    Abstract: Let (R,m) be a local ring in prime characteristics. The diagonal F-threshold (i.e, the F-threshold of m with respect to an m-primary ideal I) is known to exists as a real number in either the F-pure on the punctured spectrum case or in the standard-graded complete intersection case. In this talk, we make connections between the rationality of this invariant with that of the Hilbert-Kunz multiplicity by investigating the socle degrees distribution of the Artinian quotient of R (modulo the Frobenius powers).
  7. Some new results on local cohomology
    Gennady Lyubeznik, University of Minnesota
    Abstract: We will describe some striking recent results of our student Yi Zhang on local cohomology modules of polynomial rings in finitely many variables over a a field of characteristic p>0. Proofs, not surprisingly, involve the Frobenius morphism. There is no doubt that characteristic zero analogues of these results are true but proofs are yet to be found.
  8. F-signature and F-rational signature
    Mel Hochster, University of Michigan
    Abstract: After first surveying what is known about F-signature, which is a subtle invariant of local rings (R,m) of prime characteristic p > 0 that gives some measure of how singular they are, the talk will discuss recent results of Yongwei Yao and the speaker concerning a new notion, F-rational signature. Positivity of F-signature characterizes strongly F-regular rings, while positivity of F-rational signature characterizes F-rational rings. One key point is that if I is an m-primary ideal there is a positive real constant c(I) such that for all ideals J between I and m, the difference of the Hilbert-Kunz multiplicities of R with respect to I and J, if nonzero, is bounded below by c. A number of open questions will be discussed.
  9. Geometry of surfaces of general type
    Bangere Purnaprajna, University of Kansas
    Abstract: In this talk I will survey my recent results with my coauthors on varieties of general type with particular emphasis on the case of algebraic surfaces. The first theme will relate the deformation of canonical maps to construction of varieties of general type with prescribed invariants. The framework we develop allows us to describe some components of infinitely many moduli spaces of surfaces of general type. The second theme is to explore a higher dimensional analogue of the uniformization theorem of Riemann and Kobe, the so-called holomorphic convexity of the universal cover of a projective variety, which goes under the name of Shafarevich conjecture. Until recently, this was not known in its full generality for even surfaces fibered by genus two curves. We prove some general statements about fundamental groups of surfaces fibered by hyperelliptic curves of arbitrary genus. Examples show that this is an optimal result. As a byproduct we prove, a stronger form of Shafarevich conjecture for these surfaces, and a very attractive conjecture of Nori on fundamental groups. This also yields statements on second homotopy groups of fibered surfaces.
  10. Local theory of self maps,
    Lucien Szpiro, City University of New York
    Abstract: We will report on work with Mahdi Majidi Zolbanin and Nikita Miasnikov concerning self maps of local rings:
    1. Extension theorems a la Fakhruddin;
    2. Regularity criterium a la Kunz (and Avramov, Miller, and Iyengar);
    3. Existence of entropy a la Samuel.
  11. Multiplicities and equisingularity
    Bernd Ulrich, Purdue University
    Abstract: This talk will give a general survey of the connection between equisingularity theory and multiplicities. The last part of the lecture will be devoted to more recent results and a new notion of multiplicity, the ε-multiplicity.
    One of the goals in equisingularity theory is to devise criteria for analytic sets to be 'alike', most notably when these sets occur in a family. Ideally, such criteria only depend on numerical information about the individual members rather than the total space of the family. The numerical invariants often used are suitably dened multiplicities. Many of the known equisingularity conditions, such as Whitney's condition B or Verdier's condition W, involve limits of tangent spaces, and it was Teissier's seminal insight to relate these convergence properties to the purely algebraic concept of integral dependence of ideals. Thus he was able to show that a family of isolated hypersurface singularities is Whitney equisingular if and only if the Hilbert-Samuel multiplicity of certain Jacobian-like ideals is constant across the family. Any generalization beyond the case of isolated hypersurface singularities however, necessitates the use of Jacobian modules rather than ideals and requires new notions of multiplicities. The last part of the talk will survey such generalizations, including recent work with S. Kleiman and J. Validashti, were families of arbitrary isolated singularities are treated.