Algebraic Geometry Seminar Series

• March 7, 2012 - Marzieh Bayeh

  Bezout's Theorem
  Abstract

• March 7, 2012 - Ruhi Ahmadi

  Hirzebruch Surfaces, Part II
  Abstract

• March 14, 2012 - Gurmail Singh

  Affine Schemes
  Abstract

• March 7, 2012 - Ruhi Ahmadi

  Hirzebruch Surfaces
  Abstract

• February 29, 2012 - Ruhi Ahmadi

  Divisors and their Line Bundles
  Abstract: We explain how a line bundle $\pi: L\to M$ is related to $H^1(M,\mathcal{O}^*)$ and then the connection between divisors and line bundles. The hyperplane section bundle and universal bundle over a complex projective space are also discussed. There is a geometric construction of the universal bundle of $\mathbb P_1$ which can also be interpreted as the space resulting from the blow up of $\mathbb{C}^2$ at the origin.

• February 22, 2012 - Bruce Gilligan

  Line and Vector Bundles
  Abstract: Definition and basic properties of complex vector bundles on a smooth manifold; particulars about holomorphic line bundles on a complex manifold; some examples and their properties.

• February 15, 2012 - Bruce Gilligan

  Cohomology of Sheaves
  Abstract: Definition of the cohomology of a topological space with coefficients in a sheaf: first for the nerve of a covering, behaviour with respect to refinements, direct limits, and the exact sequence in cohomology associated to a short exact of sheaves.

• February 8, 2012 - Bruce Gilligan

  Sheaves
  Abstract: Definition and elementary properties of presheaves and sheaves on a topological space.

• February 1, 2012 - Ruhi Ahmadi

  Computing Self-intersections
  Abstract: How to compute the self-intersection of the blow-up of a point in a complex surface and finding how the self-intersection of a curve behaves when a point on that curve is blown up.

• January 25, 2012 - Bruce Gilligan

  Invariants
  Abstract: Introduction to some invariants associated to algebraic varieties.

• December 6, 2011 - Don Stanley

  Intersection Pairing
  Abstract: Definition and basics about the intersection pairing on a compact oriented 4-manifold.

• November 29, 2011 - Don Stanley

  Homology Theories
  Abstract: Definition and properties of the homology/cohomology of cell complexes and simplicial complexes.

• November 22, 2011 - Yong Liu

  Rational Maps Between Varieties
  Abstract: (Dominant) rational maps are introduced and an equivalence between the category of varieties with their dominant rational maps and the category of finitely generated field extensions of k, via the function field functor, is established. The characterizations of birational equivalence via the function fields are also discussed.

• November 15, 2011 - Yong Liu

  Morphisms Between Varieties
  Abstract: Morphisms between varieties are defined and an equivalence between the category of varieties with their morphisms and the category of finitely generated k-algebras which are also integral domains with their k-algebra homomorphisms, via the coordinate ring functor, is established.

• November 8, 2011 - Yong Liu

  Regular Functions on Affine Varieties
  Abstract: Introduction to the regular functions on affine varieties. The connection between affine varieties and its coordinate rings via the ring of regular functions, the local rings and the function field is discussed.

• November 1, 2011 - Yong Liu

  Affine Varieties And Coordinate Rings
  Abstract: Introduction to the affine n-spaces over a field k, algebraic sets and Zariski topology. Establish a 1-1 correspondence between the sets of algebraic sets in the affine n-space to the set of radical ideals in the polynomial ring of n variables over k. Affine varieties and their coordinate rings are defined. The dimensions of varieties are also discussed.

• October 25, 2011 - Jeremy Lane

  Resolving Curve Singularities
  Abstract: The strict and total transforms of plane curves by blowing up. Several examples of a resolving singularities of plane curves and Hironaka's Theorem for plane curves with singularities. An introduction to rational and birational maps, the definition of a resolution of a projective variety, and the resolution of a general cusp. Some discussion of birational maps as a composition of blow ups and blow downs with the example of the Cremona transformation.

• October 18, 2011 - Jeremy Lane

  Blow-ups
  Abstract: The product of projective spaces as a variety in a higher dimensional projective space via the Segre embedding and the construction of the blow up of projective space and affine space at the origin, with 2 diagrams.

• October 11, 2011 - Allen Herman

  Singularities
  Abstract: Singular points, n-fold singularities, dual curve, class of a curve = degree of its dual curve, calculations of dual curves.