WAGS Spring 2013

WAGS / Spring 2013

16-17 February 2013

Western
Algebraic
Geometry
Symposium

Federico Ardila
Title: Two applications of combinatorics to (tropical) enumerative geometry. Abstract: Tropical geometry is a valuable tool which translates questions in classical enumerative geometry into questions about polyhedral complexes. It is no suprise that the resulting combinatorial questions are usually rather subtle. I will discuss two instances where combinatorial techniques have provided (somewhat puzzling) new insight into classical problems: Severi degrees of toric surfaces, and double Hurwitz numbers. The first of these two projects is joint with Florian Block (UC Berkeley).

Title: Two applications of combinatorics to (tropical) enumerative geometry.

Abstract: Tropical geometry is a valuable tool which translates questions in classical enumerative geometry into questions about polyhedral complexes. It is no suprise that the resulting combinatorial questions are usually rather subtle. I will discuss two instances where combinatorial techniques have provided (somewhat puzzling) new insight into classical problems: Severi degrees of toric surfaces, and double Hurwitz numbers.

The first of these two projects is joint with Florian Block (UC Berkeley).

Noah Giansiracusa
Title: The moduli space of stable rational curves---one G_a away from toric Abstract: I'll discuss joint work with Brent Doran (ETH) in which we construct an "algebraic uniformization" of the moduli space \bar{M}_{0,n} of stable, rational n-pointed curves: we present it as a non-reductive GIT quotient of affine space by a non-linearizable action of a solvable group. This is accomplished by replacing the universal torsor with an A^1-homotopic model that encodes the global projective geometry of the moduli space in a more accesible way. We show in particular that the Cox ring of \bar{M}_{0,n} is a G_a-invariant subring of a polynomial ring, which allows it to be computed for any fixed n, in principle, by standard invariant theory algorithms, and which shows a precise sense in which \bar{M}_{0,n} is "one G_a away" from being a toric variety.

Noah Giansiracusa

Title: The moduli space of stable rational curves---one G_a away from toric

Abstract: I'll discuss joint work with Brent Doran (ETH) in which we construct an "algebraic uniformization" of the moduli space \bar{M}_{0,n} of stable, rational n-pointed curves: we present it as a non-reductive GIT quotient of affine space by a non-linearizable action of a solvable group. This is accomplished by replacing the universal torsor with an A^1-homotopic model that encodes the global projective geometry of the moduli space in a more accesible way. We show in particular that the Cox ring of \bar{M}_{0,n} is a G_a-invariant subring of a polynomial ring, which allows it to be computed for any fixed n, in principle, by standard invariant theory algorithms, and which shows a precise sense in which \bar{M}_{0,n} is "one G_a away" from being a toric variety.

Ravi Vakil
Title: The Grothendieck ring of varieties Abstract: I will give an introduction to the Grothendieck ring of varieties, giving a survey of a small portion of interesting results in the field due to various people (Kapranov, Cheah, Larsen-Lunts, Liu-Sebag, Bittner, Litt, ...).

Chenyang Xu
Title: The dual complex of singularities Abstract: Dual complex of a singularity is originally defined as the homotopy class of CW complexes which characterize how the exceptional divisors in a log resolutions intersect each other. Using MMP, for isolated singularities, we indeed find a canonical representative of the dual complex which is defined up to PL homeomorphism. The same method also answers the question that the dual complex of a klt singularity is always contractible. This is a joint work with de Fernex and Koll\'ar.

Chenyang Xu

Title: The dual complex of singularities

Abstract: Dual complex of a singularity is originally defined as the homotopy class of CW complexes which characterize how the exceptional divisors in a log resolutions intersect each other. Using MMP, for isolated singularities, we indeed find a canonical representative of the dual complex which is defined up to PL homeomorphism. The same method also answers the question that the dual complex of a klt singularity is always contractible. This is a joint work with de Fernex and Koll\'ar.

Xinyi Yuan
Title: Noether Inequalities on algebraic surfaces and arithmetic surfaces Abstract: In this talk, I will first introduce a refined Noether inequality for fibered algebraic surfaces. Then we introduce an arithmetic version of the inequality in the setting of Arakelov theory. These are joint works with Tong Zhang.

Zhiwei Yun
Title: Zeta function of a singular curve Abstract: The zeta function of a smooth curve over a finite field packages information about the number of points on it in larger and larger finite fields. It is a rational function and has a functional equation. How do we define a zeta function for a singular curve that captures not just the number of points but also information about its singularities? Does this function enjoy similar properties of the usual zeta function? What is its analog for number fields? We will answer some of these questions in the talk.

Zhiwei Yun

Title: Zeta function of a singular curve

Abstract: The zeta function of a smooth curve over a finite field packages information about the number of points on it in larger and larger finite fields. It is a rational function and has a functional equation. How do we define a zeta function for a singular curve that captures not just the number of points but also information about its singularities? Does this function enjoy similar properties of the usual zeta function? What is its analog for number fields? We will answer some of these questions in the talk.

Department of Mathematics
Harvey Mudd College

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