# Seminars & Events for Algebraic Geometry Seminar

##### Derived Equivalence and the Picard Variety

I will explain a result, joint with Mihnea Popa, saying that if two smooth projective varieties have equivalent derived categories of coherent sheaves, then their Picard varieties are isogeneous; in particular the number of independent holomorphic one-forms is a derived invariant. A consequence of this is that derived equivalent threefolds have the same Hodge numbers.

##### Gradient ideals

Zero schemes of exact 1-forms have received more attention recently as moduli spaces associated to Calabi-Yau threefolds; they are called gradient schemes or critical schemes. In this talk I will introduce the notion of "multi-gradient schemes" as an obvious generalization and explain their classification in the codimension one and monomial cases, as well as how they naturally arise as certain moduli spaces associated to varieties with globally generated canonical bundles.

##### Equivariant birational maps and resolutions of categorical quotients

If $X^{ss}$ is the set of semi-stable points for a linearized action of a reductive group on a smooth projective variety $X$ then there two procedures (Kirwan's procedure or change of linearization) for constructing a partial resolution of singularities of the categorical quotient $X^{ss}/G$. Both involve finding an equivariant birational map $\tilde{X} \to X^{ss}$ with $\tilde{X}$ smooth such that $G$ acts properly on $\tilde{X}$ and the induced map on quotients is proper and birational. A natural question to ask is whether (and to what extent) this procedure can be replicated for non-GIT quotients. We consider the problem for actions of diagonalizable groups and show that there is a simple combinatorial procedure that replicates Kirwan's construction for non-projective toric varieties.

##### A vanishing theorem in characteristic p

While the classical Kodaira vanishing theorem is false in general in characteristic $p>0$, Deligne, Illusie and Raynaud proved that it remains true under some mild (liftability and dimension) conditions. It has then been generalized in two directions: (1) Esnault and Viehweg allowed the line bundle to be somewhat less than ample and (2) Illusie covered the case with some nontrivial coefficients. We prove a common generalization of the two, which then yields a vanishing theorem of Kollár type.

##### 4-dimensional symplectic holomorphic contractions

In the talk I will consider birational projective morphisms from smooth holomorphic symplectic fourfolds into affine normal varieties. The ultimate goal will be to classify such maps. For the moment I can present special features of them and discuss some important examples.

##### Around the Tate conjecture with integral coefficients

Due to the analogy with the Hodge conjecture, it has been known for a long time that the Tate conjecture for algebraic cycles on varieties over finite fields does not hold if one considers the cycle map into \'etale cohomology with Z_\ell-coefficients. Still, some cases may be expected to hold and they have interesting consequences. We shall explain some of the negative and positive results.

##### Special Gamma, Zeta, Multizeta values and Anderson t-Motives

We will describe how the "special value theory" in function field arithmetic is an interesting mixture of very strong theorems determining all algebraic relations in some cases, emerging partial conjectural pictures in some cases, and quite wild phenomena often.

##### The tautological ring of $M_g$

I will talk about an approach to the ring generated by the kappa classes via the moduli space of stable quotients. The main new result (with A. Pixton) is a proof of a conjecture by Faber and Zagier of an elegant set of relations. Whether these are all the relations is an interesting question. I will discuss the data on both sides.

##### Restriction varieties and geometric branching rules

In representation theory, a branching rule describes the decomposition of the restriction of an irreducible representation to a subgroup. Let $i: F' \rightarrow F$ be the inclusion of a homogeneous variety in another homogeneous variety. The geometric analogue of the branching problem asks to calculate the induced map in cohomology in terms of the Schubert bases of $F$ and $F'$. In this talk, I will give a positive, geometric rule for computing the branching coefficients for the inclusion of an orthogonal flag variety in a Type-A flag variety. The geometric rule has many applications including to the restrictions of representations of $SL(n)$ to $SO(n)$, to the study of the moduli spaces of rank 2 vector bundles on hyperelliptic curves and to presentations of the cohomology ring of orthogonal flag varieties.

##### Restriction of sections for abelian schemes

I will describe work with Jason Starr in which we show that the group of sections of a family of abelian varieties over a higher dimensional base is determined by the restriction of the family to a "very general conic curve".

##### Symplectic deformation invariance of rationally connected 3-folds

##### Globally F-regular and Log Fano Varieties

Globally F-regular varieties are a class of projective varieties over a field of prime characteristic, closely related to the more well-known class of Frobenius split varieties, but more robust. Examples include Schubert and related varieties. In trying to understand their geometry, we discovered that they are very closely related to log Fano varieties. I will explain both these classes of varieties and the emerging connection between them.

##### Cluster Algebras and Quiver Grassmannians

A cluster algebra, which was introduced by Fomin and Zelevinsky, is a commutative algebra with a family of distinguished generators (the cluster variables) grouped into overlapping subsets (the clusters) which are constructed by mutations. A quiver Grassmannian is a projective variety parametrizing subrepresentations of a quiver representation with a given dimension vector. After introducing how cluster algebras are related to the Euler characteristics of quiver Grassmannians, we give explicit expressions for the Euler characteristics in the rank 2 case.

##### Kernels for categories of graded singularities

Due to work of D. Orlov, there is a strong relationship between the derived category of coherent sheaves on a hypersurface in projective space and the category of singularities of the Z-graded affine cone over the hypersurface. For a pair of graded hypersurfaces, I will describe another category of graded singularities which is equivalent to the category of functors between the pair. Combining our construction with Orlov's result and a result of B. Toën provides interesting relationships between the derived categories of some projective varieties. This is joint work with D. Favero and L. Katzarkov.

##### On the three compactifications of Siegel space

The moduli space of $A_g$ of abelian varieties has three classical toroidal compactifications: (1) perfect, (2) 2nd Voronoi, and (3) Igusa blowup, each with its own distinct geometric meaning. It is an interesting problem to understand exactly how these compactifications are related. I will show that (1) and (2) are isomorphic in a neighborhood of the image of a regular map from the Deligne-Mumford's moduli space $\bar M_g$, and that the rational map $\bar M_g \to \bar A_g$ for (3) is not regular for $g>8$. This is a joint work with Adrian Brunyate.

##### Theta Functions

In this talk we consider Weierstrass points for the linear space of meromorphic functions on a compact Riemann surface whose divisors are multiples of $\frac{1}{P_0^{\alpha}P_1..P_{g-1}}$ where the points $P_i$ are points on the surface and $\alpha$ is a positive integer for which there is no holomorphic differential whose divisor is a multiple of $P_0^{\alpha)P_1..P_{g-1}}$. Thus by the Riemann Roch theorem the dimension of the space is precisely $\alpha$. It develops that here are two different ways to define the Weierstrass points for this space. One way is to consider the Wronskian determinant of a basis for the space and to define the Weierstrass points as the zeros of the Wronskian and the weight of the Weierstrass point as the order of the zero.