These are my live-texed notes of the DAG Summer Minicourse 2020 at U Texas taught by Rok Gregoric, so all credit for knowledge goes to him and all credit for mistakes and lost-in-translations goes to me!

I'll upload the next lectures in this same page as they happen.

→ Lecture 1 August 3, 2020

0. Motivation

Let $C, C'\subseteq \mathbb{P}^2$ be two nice algebraic curves. They have an intersection class $[C].[C']$ living in the Chow ring, which can we expressed as

$$ ⁍ $$

in terms of the intersection multiplicities $\mu_p(C,C')$ at each point $p$. If $C\overline{\pitchfork}C'$ ($C$ intersects $C'$ transversally) at $p$, then $\mu_p(C, C') = 1$. In general, if they intersect non-transversally, we compute the intersection multiplicy by deforming the curves a little bit then reading off the intersection number.

There exists a closed form formula which computes this intersection multiplicity:

$$ \mu_p(C, C') = \text{dim}(\mathcal{O}{C, p}\mathop{\otimes}\limits{\mathcal{O}{\mathbb{P}^2, p}} \mathcal{O}{C', p})\\ \ \ \ \ \ = \text{dim}(\mathcal{O}{C\mathop{\times}\limits{\mathbb{P}^2}C', p}) $$

where $\text{dim}$ here indicates height. The ring $\mathcal{O}_{C\mathop{\times}{\mathbb{P}^2}C', p}$ is the local ring at $p$ of the scheme-theoretic intersection of $C$ and $C'$, which is different from the ordinary intersection in the same way that schemes differ from ordinary varieties: there may be nilpotence.

Intersection classes are not just a thing for curves. For higher dimensional subvarieties $Y, Y'\subseteq X$, we have an intersection class

$$ [Y].[Y'] = \sum_{Z\subset Y\cap Y'}\mu_Z(Y,Y') [Z] $$