#Princeton algebraic geometry seminar mod

#Princeton algebraic geometry seminar mod

The proof utilizes an interesting technique: to prove a topological fact about a complex surface we use algebraic reduction mod p and deformation theory. This makes the Craighero-Gattazzo surface the only explicitly known example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class.

This was first conjectured by Dolgachev and Werner, who proved that its fundamental group has trivial profinite completion. We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. The Craighero-Gattazzo surface is simply-connected

I will explain the motivation for this problem and discuss a recent result proving the existence of normalized volume minimizers. While Li's problem is closelyĬonnected to the notion of K-semistability, it also relates to an invariant of singularities previously explored in the work of de Fernex, Ein,Īnd Mustata. Motivated by work in Kahler-Einstein geometry, Chi Li defined the normalized volume function on the space of valuations overĪ singularity and proposed the problem of both finding and studying the minimizer of this function. I will discuss recent work (with Dori Bejleri) towards constructing various modular compactifications of spaces of elliptic surface pairs analogous to Hassett's moduli spaces of weighted stable curves.įebruary 28 (Special time/room: 2 pm, Fine 401) Moduli spaces of weighted stable elliptic surfaces This is joint work with Brendan Hassett.įebruary 21 (Special time/room: 2 pm, Fine 401) This example also provides an interesting relation in the Grothendieck ring of complex algebraic varieties. Could it be realized further through an explicit construction of birational geometry? In this talk, I will present an example where the derived equivalences of K3 surfaces are explained through Cremona transformations of P^4. In the case of K3 surfaces, this equivalence is realized as an Hodge isometry between the transcendental lattices according to Mukai and Orlov. Two varieties are called derived equivalent if their bounded derived categories of coherent sheaves are isomorphic to each other. Using some simple ideas from Berkovich analytic geometry.Ĭremona Transformations and Derived Equivalences of K3 Surfaces The ideas in my recent construction, joint with Tony Yu, of the algebra in dimension two I'll explain the conjecture, these applications, and then some of

Gives a canonical basis for every irreducible representation of a semi-simple group (withoutĭoing any representation theory!). Instances are a basis of the Cox ring of a FanoĬanonically determined by a single choice of anti-canonical divisor, one example of which In which the structure constants for the multiplication rule are given byĬounts of rational curves on the mirror. Vector space of regular functions on an affine log CY (with maximal boundary) comes withĪ canonical basis, generalizing the monomial basis on a torus, Gross, Hacking, Siebert and I conjecture that the Theta functions for affine log CY varieties