# I: Definitions and Examples 

## 1.1: Introduction

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Machinery used to study varieties:

- Various cohomology theories
- Resolutions of singularities
- Intersection theory and cycles
- Riemann-Roch theorems
- Vanishing theorems
- Linear systems (via line bundles and projective embeddings)

Varieties that arise as examples

- Grassmannians
- Flag varieties
- Veronese embeddings
- Scrolls
- Quadrics
- Cubic surfaces
- Toric varieties (of course)
- Symmetric varieties and their compactifications

Misc notes:

- Toric varieties are always rational
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- Toric varieties: normal varieties $X$ with $T\injects X$ contained as a dense open subset where the torus action $T\times T\to T$ extends to $T\times X\to X$.
- Any product of copies of $\AA^n, \PP^m$ are toric.
- $S_\sigma$ is a finitely-generated semigroup, so $\CC[S_\sigma] \in \alg{\CC}^\fg$ corresponds to an affine variety $U_\sigma \da \spec \CC[S_\sigma]$.
- If $\tau \leq \sigma$ is a face then there is a map of affine varieties $U_\tau \to U_\sigma$ where $U_\tau = D(u_\tau)$ is a principal open subset given by the function $u_\tau$ picked such that $\tau = \sigma \intersect u_\tau^\perp$, so $u_\tau$ corresponds to the orthogonal normal vector for the wall $\tau$.
- These glue to a variety $X(\Delta)$.
- Smaller cones correspond to smaller open subsets.
- The geometry in $N$ is nicer than that in $M$, usually.
- Rays $\rho$ correspond to curves $D_\rho$.

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:::{.exercise title="?"}
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- Show $F_a\to \PP^1$ is isomorphic to $\PP(\OO(a) \oplus \OO(1))$.
- Let $\tau$ be the ray through $e_2$ in $F_a$ and show $D_\tau^2 = -a$.
- Show that the normal bundle to $D_\tau \injects F_a$ is $\OO(-a)$.

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## 1.2: Convex Polyhedral Cones


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- **Convex polyhedral cones**: generated by vectors $\sigma = \RR_{\geq 0}\gens{v_1,\cdots, v_n}$.
  Can take minimal vectors along these rays, say $\rho_i$.

![](Reading%20Notes/9_Fulton/figures/2022-10-18_20-35-45.png)

- $\dim \sigma \da \dim_\RR \RR \sigma \da \dim_\RR (-\sigma + \sigma)$
- $(\sigma\dual)\dual = \sigma$, which follows from a general theorem: for $\sigma$ a convex polyhedral cone and $v\not\in \sigma$, there is some support vector $u_v\in \sigma\dual$ such that $\inp{u}{v} < 0$.
  I.e. $v$ is on the negative side of some hyperplane defined in $\sigma\dual$.
- Faces are again convex polyhedral cones, faces are closed under intersections and taking further faces.
- If $\sigma$ spans $V$ and $\tau$ is a facet, there is a unique $u_\tau\in \sigma\dual$ such that $\tau = \sigma \intersect u_\tau^\perp$; this defines an equation for the hyperplane $H_\tau$ spanned by $\tau$.
- If $\sigma$ spans $V$ and $\sigma\neq V$, then $\sigma = \intersect_{\tau\in \Delta} H_\tau^+$, the intersection of positive half-spaces.
  - An alternative presentation: picking $u_1,\cdots, u_t$ generators of $\sigma\dual$, one has $\sigma = \ts{v\in N \st \inp{u_1}{v} \geq 0, \cdots, \inp{u_t}{v}\geq 0}$. 
- If $\tau \leq \sigma$ then $\sigma\dual \intersect \tau\dual \leq \sigma\dual$ and $\dim \tau = \codim(\sigma\dual \intersect \tau\dual)$, so the faces of $\sigma, \sigma\dual$ biject contravariantly.
- If $\tau = \sigma \intersect u_\tau^\perp$ then $S_\tau = S_\sigma + \NN\gens{-u_\tau}$.
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