Student Algebraic Geometry Seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
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Friday
February 09, 2018

2:30 PM - 3:30 PM
Math Tower 5-127
Ben Wu, Stony Brook University
Pluricanonical forms and Gorenstein varieties

For a smooth variety, the canonical bundle is defined to be the line bundle which is the determinant of the cotangent bundle. A canonical divisor is a divisor whose associated line bundle is the canonical bundle. For normal varieties, we still have a notion of canonical divisor despite the presence of singularities. However, the sheaf associated to this canonical divisor may fail to be locally free, i.e. is not a line bundle. When the sheaf associated to some multiple of the canonical divisor on a normal variety is locally free, the variety is called Q-Gorenstein. In this talk, we will discuss how to define the canonical divisor on normal varieties and its associated sheaf and will discuss examples and non-examples of Q-Gorenstein varieties.


Friday
February 16, 2018

2:30 PM - 3:30 PM
Math Tower 5-127
Ying Hong Tham, Stony Brook University
Canonical and terminal singularities with examples

In this talk, we will pick up the discussion after last week's introduction to Q-Gorenstein varieties. We will introduce the notion of canonical and terminal singularities on such varieties, giving some motivating examples to illustrate some of the general behaviors exhibited.


Friday
February 23, 2018

2:30 PM - 3:30 PM
Math Tower 5-127
Sasha Victorova, Stony Brook University
Canonical models, minimal models and their singularities

In this talk we will run with the definitions of canonical and terminal singularities from last week and use them to give definitions of canonical and minimal models of (quasi-)projective varieties. As usual we will turn to examples to motivate the main ideas.


Friday
March 09, 2018

2:30 PM - 3:30 PM
Math Tower 5-127
Yoonjoo Kim, Stony Brook University
Finite generation of the canonical ring and existence of the canonical model

Today, I will talk about the proof of the following theorem, which initiated people to study the canonical ring of a variety:
Theorem (M. Reid, 1980) Let V be a smooth projective variety of general type. If a canonical model of V exists, it has to be the Proj of the canonical ring. Moreover, existence a of canonical model is equivalent to the finite generation of the canonical ring.
In 2009, Birkar-Cascini-Hacon-McKernan finally proved the existence of a canonical model, so now we know the canonical model is exactly the Proj of the canonical ring.


Friday
March 30, 2018

2:30 PM - 3:30 PM
Math Tower 5-127
John Sheridan, Stony Brook University
Reduction of canonical singularities - codimension 2 analysis

We will recall what we have done in the seminar so far leading us most recently to the proof of Reid's theorem on canonical models/rings. We contrast this with the state of the art in dimension 2 and then broadly indicate the diverging relationship between smooth, minimal and canonical models in dimensions 3 and above. Primarily then, we take the first step in a path towards reduction of canonical singularities to a class of ("hyperquotient") singularities that, in dimension 3, has been largely classified by Mori. Where possible, we will discuss examples.


Friday
April 20, 2018

2:30 PM - 3:30 PM
Math Tower 5-127
Frederik Benirschke, Stony Brook University
Rationality of canonical singularities

In this talk, we will continue with the process of reducing the study of canonical singularities to the special class of hyperquotient singularities as outlined by Reid. Today we indicate a sufficient criterion for rationality of higher dimensional singularities and show that this is achieved by canonical ones. We will also have some pertinent discussion of the Cohen-Macaulay property of a variety in the present context.


Friday
April 27, 2018

2:30 PM - 3:30 PM
Math Tower 5-127
Lisa Marquand, Stony Brook University
Hyperplane sections of canonical singularities

In a recent talk, we showed that away from a small closed subset, canonical singularities on a Q-Gorenstein variety X are products by dimension 2 RDPs. An immediate corollary was that a general hyperplane section of X also has canonical singularities. In this talk, we ask for a little more regularity on X (that it is properly Gorenstein - a situation we can reduce to by previous talks) and show in this setting that when we fix a canonically singular point p, the general hyperplane section through p has either rational or elliptic singularities. Intuitively these are worse singularities than canonical, but still relatively well-behaved.


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