# Algebraic Geometry Seminar

### Spring 2013 — Tuesdays 3:30-4:30, LCB 215 (note new location)

 Date Speaker Title — click for abstract (if available) August 21 No Seminar August 28 Yi Zhu University of Utah Finding very free rational curves A rational curve on a smooth algebraic variety is very free if the pullback of the tangent bundle is positive. This notion is the key to the theory of rational curves and has various applications both in geometry and in arithmetics. In characteristic zero, the general theory of Kollar-Miyaoka-Mori predicts the existence of such curves on Fano varieties (or more generally, on rationally connected varieties). However, constructing very free curves is usually difficult, especially in arbitrary characteristic. In this talk, first I will survey basic results which motivate the search of very free curves. Then I will focus on several examples where we can construct them concretely. September 4 Eric Katz University of Waterloo Lifting Tropical Curves and Linear Systems on Graphs Tropicalization is a procedure for associating a polyhedral complex to a subvariety of an algebraic torus. We explain the method of tropicalization and study the question of which graphs arise from tropicalizing algebraic curves. By applying Baker's technique of specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition is phrased in terms of the harmonic theory of graphs, reproduces the known necessary conditions, and also gives new conditions. Moreover, our method gives a combinatorial way of thinking about the deformation theory of algebraic varieties. September 11 Karl Schwede Penn State A generic restriction theorem for test ideals Suppose that f : X \to S is a flat family over a smooth variety of characteristic p > 0. In the same setting in characteristic zero, the multiplier ideal of X restricts to the multipier ideal of the fibers for most fibers X_s. There is no hope that the same statement can hold in characteristic p > 0 (either for multiplier ideals, or their analogs test ideals) because it might happen that even if X is smooth, the fibers of f might not even be reduced, and even if they are reduced, might have horrendus singularities. In this talk I will discuss a way to correct for this phenomenon, and thus obtain a generic restriction theorem for test ideals in positive characteristic. This is joint work with Zsolt Patakfalvi and Wenliang Zhang. September 183pm - 4pmin JTB 120 Xiaowei Wang Rutgers, the State University of New Jersey Hilbert-Mumford criterion for nodal curve. In this joint work with Jun Li, we prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. 4:30pm - 5:30pmin LCB 225Special Algebraic Geometry Seminar Herb Clemens The Ohio State University The holomorphic height pairing In joint work with Mirel Caibar we show that the classical height pairing between algebraic (n-1)-cycles on a (2n-1)-dimensional complex projective manifold X is the imaginary part of a natural (multivalued) bi-holomorphic function on components of the Hilbert scheme of X. This pairing is intimately related to the Abel-Jacobi image of the respective cycles. Furthermore this pairing can be extended to integral currents whose support is a real (2n-2)-dimensional oriented submanifold of X. Properties and potential applications of the extended pairing will be presented. September 25 Luigi Lombardi University of Illinois at Chicago Derived equivalences of irregular varieties and Hochschild homology In this talk I will describe the behavior under derived equivalence of a twisted version of Hochschild homology. This result is then applied to study the derived invariance of cohomological support loci, fibrations onto curves, the Albanese dimension, and certain Hodge numbers of special classes of irregular varieties. October 2 Takashi Kimura Boston University Power operations in inertial K-theory and some applications If X is a smooth variety with a proper action of an algebraic group G, its equivariant K-theory ring $K_G(X)$ possesses power (or Adams) operations (and associated lambda ring operations) which are compatible with the Chern character and Chern classes. But $K_G(X)$ is a subring of the equivariant K-theory $K_G(IX)$ of the inertial variety IX endowed with a so-called inertial product. The prototypical example of such an inertial K-theory ring is the K-theoretic version of the Chen-Ruan orbifold cohomology. We show that under certain conditions, the inertial K-theory ring admits inertial generalizations of power operations, Chern classes, and the Chern character. The power operations are then used to compare the virtual K-theory ring of P(1,2) and P(1,3) with the K-theory of a crepant resolution. This is joint work with with D. Edidin and T. Jarvis. October 9 Fall Break October 16 Olivier Benoist École Normale Supérieure Moduli spaces of smooth complete intersections In this talk, I will consider the moduli problem for smooth complete intersections in projective space. I will show the existence of a separated coarse moduli space. This coarse moduli space will be shown to be quasi-projective in some particular cases (for instance for codimension 2 Fano complete intersections). October 18 3pm - 4pm (Note different date and time) Paul Hacking University of Massachusetts Amherst Explicit 3-fold flips We complete the explicit description of 3-fold flips studied by Kollár and Mori. The new cases occur in the minimal model program for a one parameter family of surfaces, which is used to describe compact moduli spaces of surfaces. The classification is understood in terms of a universal family of K-negative surfaces with an interesting combinatorial structure. This is joint work with Jenia Tevelev and Giancarlo Urzua. October 23 Zhiyu Tian California Institute of Technology Extremal rays, Gromov-Witten invariants, and rationally connected varieties This work is motivated by Kollár's conjecture on symplectic deformation invariance of rational connectedness and "symplectic birational geometry". I will discuss some simple observations which seem to suggest that extremal rays of the Mori cone deform together with the deformation of the variety as a symplectic manifold. In some special cases, one can use these observations to prove that the maximal rationally connected quotient of a variety is a symplectic deformation invariant, a stronger version of Kollár's original conjecture. October 30 Zhixian Zhu University of Michigan Divisorial valuations via arcs in positive characteristic When X is a smooth complex variety, it was shown by Ein, Lazarsfeld and Mustata that there is a general correspondence between cylinders in the space of arcs of X and divisorial valuations of the function field of X. Via this correspondence, the codimension of the cylinders corresponds to the log discrepancy of the divisorial valuation. The use of log resolutions in their proof restricted the result to ground field of characteristic zero. In this talk, I'll show this correspondence holds in arbitrary characteristic. In particular, we have Mustata's formula, relating the log canonical threshold of a pair to the asymptotic behavior of the dimensions of the jet schemes, also available in positive characteristic. This has interesting applications, for example, to an inequality between log canonical threshold and F-pure threshold and to a version of inversion of adjunction in positivity characteristic. November 6 Colleen Robles Texas A&M University Homological flexibility of Schubert classes in cominuscule rational homogeneous varieties The Schubert subvarieties of a rational homogeneous variety X are distinguished by the fact that their homology classes form an additive basis of the integer homology of X. It is then natural to ask: does a Schubert classes admit any other algebraic representatives (aside from the Schubert variety)? It is a remarkable consequence of Kostant's work on Lie algebra homology that, in the the case that X is cominuscule -- equivalently, X admits the structure of a compact Hermitian symmetric space (eg. X is a complex Grassmannian) -- the algebraic representatives of a Schubert class are solutions of a system of differential equations. This allows us to apply differential geometric techniques to the problem. I will discuss this, and related questions, and the essential role played by representation theory in determining both rigidity (the Schubert varieties are the only algebraic representatives) and flexibility. November 13 Jason Starr Stony Brook University Rational points of varieties over global function fields This is joint work with Chenyang Xu. Combining work of Esnault with work of de Jong - He - Starr on "rational simple connectedness", we prove the existence of rational points of many varieties defined over global function fields, i.e., function fields of curves over finite fields. In this way we get uniform proofs and extensions of (1) Lang's theorem that every global function field $K$ is $C_2$, (2) a theorem of Brauer - Hasse - Noether that the period equals the index for every division algebra over $K$, and (3) the "split case" of Harder's proof of Serre's Conjecture II over $K$. We also get upper bounds on the heights of our rational points, independent of the characteristic. November 20 Bhargav Bhatt University of Michigan p-adic derived de Rham cohomology A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine’s conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry — specifically, derived de Rham cohomology — and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way. November 27 Mark Shoemaker University of Michigan A Mirror Theorem for the Mirror Quintic (Joint w/ Y.P. Lee) The celebrated Mirror Theorem of Givental and Lian-Liu-Yau states that the A model (quantum cohomology, rational curve counting) of the Fermat quintic threefold is equivalent to the B model (complex deformations, period integrals) of its mirror dual, the mirror quintic orbifold. In order for mirror symmetry to be a true duality however, one must also show that the B model of the Fermat quintic is equivalent to the A model of the mirror quintic. We prove such an equivalence by relating the orbifold Gromov-Witten theory of the mirror quintic to period integrals over a one parameter deformation of the Fermat quintic. This involves new calculations in orbifold Gromov-Witten theory. December 4 János Kollár Princeton University The dual complex of singularities