Date 
Topic 
Speaker 
Abstract and references 
09/27/2017 
Weighted Projective spaces and weighted blowups 
Mingyi 
Abstract
I was studying elliptic surface singularities and notice a way of getting minimal resolution of these singular surface by weighted blow ups. Also, after reading several introduction notes, there are many hyper surfaces and complete intersection sub variety in weighted projective space that are good examples in algebraic geometry. For example, we can find many Del Pezzo surfaces as hypersurfaces in weighted projective spaces. I will show some easy computation and some pathology of weighted projective varieties.
References:
Miles Reid, Graded rings and varieties in weighted projective space, 2002
J. Kollar, K. Smith and A. Corti, Rational and nearly rational varieties, 2004, 3.48, 6.38
Igor Dolgachev, Weighted projective varieties, 1982.
Anthony IanoFlecher, Working with weighted complete intersection, in the book Explicit binational geometry of 3folds.

10/03/2017 
Fujita type conjectures 
Yagna 
Abstract
I One of the most natural questions in algebraic geometry is whether there is an effective bound on twists of ample line bundles to obtain global generation of coherent sheaves. Fujita, in 1985, conjectured that the bound for the canonical bundle depends only on the dimension of the variety. Despite many important breakthroughs by Reider, Ein, Lazarsfels, Kawamata et al., the conjecture remains unproved as of today. We will walk through this history and discuss some recent developments on a relative analog of Fujita's conjecture.
References: Lazarsfeld's Positivity I, Chapter 4.1, Kawamata covers. 
10/10/2017 
Period Maps and period domains 
Mingyi 
Abstract
I will give examples to show how period domain parametrizes the possible polarized Hodge structures in the cohomology of a given smooth projective variety. We will see KodairaSpencer map, Griifiths' period map and its holomorphicity and transversality.
References: Voison's book and J. Carlson's book Period mappings and period domains.

10/17/2017 
Mixed Hodge Structures, some examples 
Sebastián 
Abstract
I will talk about Mixed Hodge structures in the cohomologies of algebraic varieties. I will briefly discuss the limit Mixed Hodge structure.
References: Hodge Theory  Cattani, Zein, Griffiths, Tráng et al. 
10/24/2017 
No Seminar 

Algebraic Geometry Seminar conflict 
10/31/2017 
Analytic aspects of multiplier ideal sheaves 
Greg 
Abstract
We discuss multiplier ideal sheaves from the analytic perspective with an emphasis on examples. As an application we will discuss the proof of the Nadel vanishing theorem using Hormander's L^2 estimate for the dbarequation.

11/07/2017 
Some examples in derived categories of coherent sheaves. 
Grisha P. 
Abstract
We will discuss what derived category of a manifold knows about a manifold itself, and what it doesn't. I wanted to talk about BondalOrlov theorem, which says that in the case of ample or antiample canonical bundle the category knows more or less everything, and calculate the category explicitly in the case of projective space.
Reference: Huybrechts, FourierMukai transform in algebraic geometry (Reading through Chapter 1 is highly encouraged). 
11/14/2017 
Fourier Mukai Transform in algebraic geometry 
Grisha K. 
Abstract
FourierMukai transofrm is one of the fundamental tools in (derived) algebraic geometry. I will try to disuss this theory from the point of view of correspondences, sketch the proof of the main statement and discuss several possible applications.
References: D. Huybrechts, FourierMukai Transforms in Algebraic Geometry
D. Gaitsgory and N. Rozenblyum, A Study in Derived Algebraic Geometry
D. BenZvi, J. Francis, D. Nadler, Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry

11/28/2017 
Cohomology in characteristic p 
Paul 
Abstract
What good cohomology theories are there for varieties X in characteristic p? One candidate is the de Rham cohomology of a smooth lift of X to characteristic 0, which is surprisingly independent of the choice of lift. This cohomology theory admits an intrinsic definition  the crystalline cohomology of X  and connects familiar characteristiczero phenomena with phenomena only seen in characteristic p. After defining crystalline cohomology and discussing its properties, we'll compute several examples for curves and surfaces. This will in part serve as an excuse to visit some of the peculiarities of surfaces in positive characteristic.
References:
Berthelot and Ogus, Crystalline Cohomology (clear, booklength treatment)
Katz, "Travaux de Dwork" (in English) and "Slope filtration of Fcrystals"; Mazur, "Frobenius and the Hodge filtration" (motivation from the point of view of counting points on characteristic p varieties, and discussion of the sort of objects crystalline cohomology is valued in)
Illusie, "Complexe de deRhamWitt et cohomologie cristalline" (150 pages and in French but defines a spectral sequence used to compute cohomology and resolves Katz's conjecture; the last section has several interesting examples); see also these notes
Bhatt, "Torsion in the crystalline cohomology of singular varieties" (a more recent computation)
BhattMorrowScholze, "Integral padic Hodge theory" and the shorter announcement (deep theorem comparing various cohomology theories in mixed characteristic)
BombieriMumford, "Enriques' classification of surfaces in characteristic p IIII" (classification of surfaces)
