Topics in algebraic and analytic geometry

Advanced Topics in Algebraic Geometry

Outline of talk: My goal is to introduce a few more advanced topics in algebraic geometry but not to go into too much detail. This will be a survey of (1) Elimination Theory and review from last time (BRIEFLY) (a) History and goals (b) Geometric Extension Theorem (2) Invariant Theory (3) Dimension Theory (a) Krull Dimension (b) Hilbert Polynomial (c) Dimension of a variety (4) Syzygies (a) Defi...

Topics in Enumerative Algebraic Geometry Lecture 19

T : C −−−−→ R+  yM integer matrix T r ←−−−− R Then the toric manifold X is defined to be J(ω)/T r , where ω is a point in K, an open cone in R (please refer to previous lectures). Assume that X is smooth, i. e. T r action on J(ω) is free, we have: H∗(X) = H∗ T r(J (ω)) We first notice that J(ω) is T-equivariantly homotopic to J(K) where J(K) = C \ ∪ (Coordinate subspaces which miss K under J) .

Topics in Enumerative Algebraic Geometry Lecture 13

1.2. Representation of Z. Recall that we were about to construct a representation of Ug′ in differential operators on the maximal torus T of G which, when restricted to Z, will yield, after some modification, a set of commuting differential operators satisfying Kim’s lemma. Their symbols will then give us the relations in the quantum cohomology of X. These operators are the integrals of the qua...

Topics in Enumerative Algebraic Geometry Lecture 28

Topics in Enumerative Algebraic Geometry Lecture 23