Harvard/MIT Algebraic Geometry Seminar

Spring 2013
Tuesdays at 3 pm

The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT (2-139) and Harvard (Science Center 507).

• May 142013 SC 507Harvard Larry Guth (MIT)
  Applications of ruled surfaces in combinatorial geometry abstract±   (pdf) We explain the connection between classical results in ruled surface theory and recent work in combinatorial geometry.
  In combinatorial geometry, we will explain the following result. Suppose that we have L lines in R³. Unless the lines cluster into a small number of planes and degree 2 algebraic surfaces, then the number of intersection points of the lines is at most C L^3/2. (This theorem is from joint work of Nets Katz and myself.)
  The proof uses results about ruled surfaces, such as the following: If an algebraic surface in R³ contains two lines through every point, then the surface is a union of planes and degree 2 surfaces. Such a surface is called doubly ruled.
  The main goal is to explain the connection between the combinatorial setting and the algebraic setting. The combinatorial result above is like a discrete version of the algebraic result. The intersection points of the lines are like the `points of a discrete surface'. This `discrete surface' has two lines through every point. It is a discrete version of a doubly ruled surface. If there are a lot of intersection points, then we are able to conclude that the discrete surface is a union of planes and degree 2 surfaces, as in the algebraic case.

• Feb 122013 2-139MIT Thomas Scanlon (UC Berkeley)
  Invariant varieties for polynomial dynamical systems abstract±   (pdf) We describe the possible invariant varieties for algebraic dynamical systems f : Aⁿ_C → Aⁿ_C of the form (x₁,...,x_n) → (f₁(x₁),...,f_n(x_n)) where each f_i is a univariate polynomial. In particular, we show that unless the polynomials come from an action of an algebraic group, the invariant varieties are all (components of) varieties defined by equations involving only two variables. Moreover, such invariant curves come from the obvious sources (for example, the curve defined by x₂ = g(x₁) is invariant under the action of (g,g)). A major component of our work is the correct enunciation of the ideas of "coming from" an algebraic group action or from an "obvious source."
  The ideas behind our classification originate in the model theory of difference equations, but the proofs pass through elementary algebraic geometry and old theorems of Ritt on compositional polynomial identities. As corollaries of the description of the invariant varieties, we deduce some cases of conjectures of Shou-Wu Zhang on dynamical analogues of classical theorems in Diophantine geometry.
  [This is a report on joint work with Alice Medvedev.]
• Feb 192013 SC 507Harvard Bernd Sturmfels (UC Berkeley)
  A Hilbert Scheme in Computer Vision abstract±   (pdf) Multiview geometry, the study of two-dimensional images of three-dimensional scenes, is foundational for computer vision. We present work with Chris Aholt and Rekha Thomas on the equations that characterize images taken by n cameras. Our varieties are threefolds that vary in a family of dimension 11n-15 when the cameras are moving. Toric geometry and Hilbert schemes are used to characterize degenerations of camera positions.
• Feb 262013 SC 507Harvard No seminar
   
• Mar 52013 2-139MIT Jørgen Vold Rennemo (Imperial College London)
  The Homology of Hilbert Schemes of a Locally Planar Curve abstract±   (pdf) Associated with an algebraic curve C are the Hilbert schemes C^[n] parametrising length n subschemes of C. If C is smooth, C^[n] is the n^th symmetric product of C, hence topologically simple, but if C is singular, the topology of C^[n] depends on the singularities of C. When C has locally planar singularities, recent work of Maulik-Yun and Migliorini-Shende shows that the rational homology groups of the C^[n] are explicitly determined by a certain filtration on the homology of the compactified Jacobian of C. I will explain this result and a new proof.
• Mar 122013 SC 507Harvard Brian Lehmann (Rice)
  Big cycles and volume functions abstract±   (pdf) The volume of a divisor is an important invariant measuring the "positivity" of its numerical class. I will discuss an analogous construction for cycles of arbitrary codimension. In particular, this yields geometric characterizations of big cycle classes modeled on the well-known criteria for divisor and curve classes.
• Mar 192013 2-139MIT No seminar
  Harvard spring break
• Mar 262013 SC 507Harvard Noah Giansiracusa (UC Berkeley)
  Tropical scheme theory abstract±   (pdf) I'll discuss joint work with J.H. Giansiracusa (Swansea) in which we study scheme theory over the tropical semiring T, using the notion of semiring schemes provided by Toen-Vaquie, Durov, or Lorscheid. We define tropical hypersurfaces in this setting and a tropicalization functor that sends closed subschemes of a toric variety over a field with non-archimedean valuation to closed subschemes of the corresponding toric variety over T. Upon passing to the set of T-valued points this yields Payne's extended tropicalization functor. We prove that the Hilbert function of any projective subscheme is preserved by our tropicalization functor, so the scheme-theoretic foundations developed here reveal a hidden flatness in the degeneration sending a variety to its tropical skeleton.
• Apr 22013 2-139MIT No seminar
   
• Apr 92013 SC 507Harvard Mike Roth (Queen's University)
  Roth's theorem for arbitrary varieties abstract±   (pdf) If X is a variety of general type defined over a number field k, then the Bombieri-Lang conjecture predicts that the k-rational points of X are not Zariski dense. The conjecture is a prediction that a global condition on the canonical bundle (that it is ''generically positive'') implies a global condition about rational points. By a well-established principle in geometry we should look for local influence of positivity on the accumulation of rational points. To do that we need measures of both these local phenomena.
  Let L be an ample line bundle on X, and x ∈ X(k̄). The central theme of the talk is the interrelations between α_x(L), an invariant measuring the accumulation of rational points around x as gauged by L, and the Seshadri constant ε_x(L), measuring the local positivity of L near x. In particular, the classic approximation theorem of K.F. Roth on P¹ generalizes as an inequality between α_x and ε_x valid for all projective varieties.
  This is joint work with David McKinnon.
• Apr 162013 2-139MIT  
  Patriots' day (MIT holiday)
• Apr 232013 SC 507Harvard François Charles
  On the twisted Lang-Weil estimates abstract±   (pdf) Given an algebraic variety over a finite field, the classical Lang-Weil estimates compute the number of intersection point of the diagonal with the graph of Frobenius. Hrushovski has proven a twisted version of these estimates, replacing the diagonal by a more general correspondence. While Hrushovski's proof relies on non-classical objects such as difference schemes, this has proven to yield a number of applications to algebraic geometry. In this talk, we will explain a self-contained intersection-theoretic proof of the twisted Lang-Weil estimates. This is joint work with Damien Rössler.
• Apr 302013 No seminar
   
• May 72013 2-139MIT Ivan Loseu (Northeastern)
  Classification of Procesi bundles abstract±   (pdf) A Procesi bundle is a vector bundle on the Hilbert scheme of n points on the plane. It was first constructed by Haiman who used it to prove the Schur positivity for Macdonald polynomials. This bundle also provides a derived McKay equivalence for the Hilbert scheme. I will basically take the latter for an axiomatic description of a Procesi bundle. I will show that there are exactly two bundles with these properties: Haiman's and its dual. Time permittingI will also discuss an extension of this results to other symplectic resolution and a relation between the Procesi bundles and the tautological bundle conjectured by Haiman. The proofs are based on the study of Symplectic reflection algebras.
• May 142013 SC 507Harvard Larry Guth (MIT)
  Applications of ruled surfaces in combinatorial geometry abstract±   (pdf) We explain the connection between classical results in ruled surface theory and recent work in combinatorial geometry. In combinatorial geometry, we will explain the following result. Suppose that we have L lines in R³. Unless the lines cluster into a small number of planes and degree 2 algebraic surfaces, then the number of intersection points of the lines is at most C L^3/2. (This theorem is from joint work of Nets Katz and myself.) The proof uses results about ruled surfaces, such as the following: If an algebraic surface in R³ contains two lines through every point, then the surface is a union of planes and degree 2 surfaces. Such a surface is called doubly ruled. The main goal is to explain the connection between the combinatorial setting and the algebraic setting. The combinatorial result above is like a discrete version of the algebraic result. The intersection points of the lines are like the `points of a discrete surface'. This `discrete surface' has two lines through every point. It is a discrete version of a doubly ruled surface. If there are a lot of intersection points, then we are able to conclude that the discrete surface is a union of planes and degree 2 surfaces, as in the algebraic case.

This seminar is organized by Joe Harris (Harvard), James McKernan (MIT), Yu-Jong Tzeng (Harvard), Vivek Shende (MIT), and Melody Chan (Harvard). Check the archives for talks from previous semesters. This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.