❕*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

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