|Date: Friday, March 30, 2018
Location: 3096 East Hall (3:10 PM to 4:00 PM)
Title: Serre's Intersection Formula and Derived Nonsense
Abstract: Given two smooth, plane algebraic curves of degree m and n, a classical theorem from Bezout tells us that if the curves intersect transversally, then the intersection has exactly mn points. If we want this formula to remain valid for non-transverse intersections, then we must count points with multiplicity by considering the non-reduced scheme structure on the intersection. However, there are still situations in which considering the scheme theoretic intersection gives the wrong answer-for example, when both curves we are trying to intersect are the same.
According to Serre's intersection formula, the correct intersection multiplicity is given by some alternating sum that involves higher Tor groups. In this talk, we will try to explain why looking at the higher Tor groups is a natural thing to do. In order to do this, we will need to move to the world of derived algebraic geometry. In this context, the higher Tor groups will naturally appear as some sort of higher nilpotent in the derived intersection. We hope to unveil much of the mystery surrounding the word "derive", by motivating the formalism as a way to approximate poorly behaved objects by good ones.
This talk will be accessible to anyone who is taking or has taken 632.
Speaker: Attilio Castano