# Singularities and Compactness 

## 2.1

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- Any cone $\sigma\in \Sigma$ has a distinguished point $x_\sigma$ corresponding to $\Hom_{\semigroup}(S_\sigma, \CC)$ where $u\mapsto \chi_{u\in \sigma^\perp}$.
  - Note $S_\sigma \da \sigma\dual \intersect M$.
- Define $A_\sigma \da \CC[S_\sigma]$.
- Finding singular points:
  - Easy case: $\sigma$ spans $N_\RR$ so $\sigma^\perp = 0$; consider $\mfm \in \mspec A_\sigma$ be the maximal ideal at $x_\sigma$, then $\mfm = \gens{\chi^u \st u\in S_ \sigma}$ and $\mfm^2 = \gens{\chi^u \st u \in S_\sigma\smz + S_\sigma\smz}$, so $\T_{x_\sigma}\dual U_\sigma = \mfm/\mfm^2 = \ts{\chi^u \st u\not \in S_{\sigma}\smz + S_\sigma\smz}$, i.e. "primitive" elements $u$ which are not the sums of two other vectors in $S_\sigma\smz$.
  - Nonsingular implies $\dim U_\sigma = n$, so $\sigma\dual$ has $\leq n$ edges since each minimal ray generator yields a primitive $u$ above. Also implies minimal edge generators must generate $S_\sigma$, thus must be a basis for $M$, so $\sigma$ must be a basis for $N$ and $U_\sigma \cong \AA^n$.
- **Characterization of smoothness**: $U_\sigma$ is smooth iff $\sigma$ is generated by a subset of a lattice basis for $N$, in which case $U_\sigma \cong \AA^k \times \GG_m^{n-k}$.

- All toric varieties are normal since each $A_\sigma$ is integrally closed.
  - If $\sigma = \gens{v_1,\cdots, v_r}$ then $\sigma\dual = \intersect_{i=1}^r \tau_i\dual$ where $\tau_i$ is the ray along $v_i$.
    Thus $A_\sigma = \intersect A_{\tau_i}$, each of which is isomorphic to $\CC[x_1, x_2^{\pm 1}, \cdots, x_n^{\pm 1}$ which is integrally closed.
- All toric varieties are **Cohen-Macaulay**: each local ring $R$ has depth $n$, i.e. contains a regular sequence of length $n = \dim R$.
- All vector bundles on affine toric varieties are trivial, equivalently all projective modules over $A_\sigma$ are free.

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## 2.2


:::{.remark}

- An example: $\Sigma = \Cone(me_1-e_2, e_2)$.
  Then $A_\sigma = \CC[x, xy, xy^2,\cdots,xy^m] = \CC[u^m u^{m-1}v,\cdots, uv^{m-1}, v^m]$ and $U_\sigma$ is the cone over the rational normal curve of degree $m$.
  - Note $A_\sigma = \CC[u,v]^{\mu_m}$ is the ring of invariants under the diagonal action $\zeta.\tv{u, v} = \tv{\zeta u, \zeta v}$.
- If $\Sigma$ is simplicial, then $X_\Sigma$ is at worst an orbifold.
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## 2.3


:::{.remark}
- $\Hom_{\Alg\Grp}(\GG_m, \GG_m) = \ZZ$ using $n\mapsto (z\mapsto z^n)$.
- Cocharacters:
  - Pick a basis for $N$ to get $\Hom(\GG_m, T_N) = \Hom(\ZZ, N) = N$, then every cocharacter $\lambda \in \Hom(\GG_m, T_N)$ is given by a unique $v\in N$, so denote it $\lambda_v$.
  Then $\lambda_v(z)\in T_N = \Hom(M, \GG_m)$ for any $z\in \CCstar$, so
  \[
  u\in M \implies \lambda_v(z)(u) = \chi^u(\lambda_v(z))= z^{\inp u v}
  .\]

- Characters: $\chi \in \Hom(T_n, \GG_m) = \Hom(N, \ZZ) = M$ is given by a unique $u\in M$ and can be identified with $\chii^u \in \CC[M] = H^0(T_N, \OO_{T_N}\units)$.

- $\lim_{z\to 0}\lambda_v(z) = \lim_{z\to 0} \tv{z^{m_1},\cdots, z^{m_n}} \in U_\sigma \iff m_i \geq 0$ for all $i$, and if $U_\sigma = \AA^k\times \GG_m^{n-k}$, $m_i = 0$ for $i > k$.
This happens iff $v\in \sigma$, and the limit is $\tv{\delta_1,\cdots, \delta_n}$ where $\delta_i = 1\iff m_i = 0$ and $\delta_i = 0\iff m_i > 0$; each of which is a distinguished point $x_\tau$ for some face $\tau$ of $\sigma$.
- Summary: $v\in \abs{\Sigma}$ and $v\in \tau\interior$ then $\lim_{z\to 0} \lambda_v(z) = x_\tau$, and the limit does not exist for $v\not\in\abs{\Sigma}$.

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## 2.4


:::{.remark}

- Recall $X$ is compact in the Euclidean topology iff it is complete/proper in the Zariski topology, i.e. the map to a point is proper.
- $X_\Sigma$ is compact iff $\abs{\Sigma} = N_\RR$, i.e. $\Sigma$ is complete.
- Any morphism of lattices $\phi:N\to N'$ inducing a map of fans $\Sigma\to \Sigma'$ defines a morphism $X_{\Sigma}\to X_{\Sigma'}$ which is proper iff $\phi\inv(\abs{\Sigma'}) = \abs{\Sigma}$.
  Thus $X_\Sigma$ is compact iff $\phi: N\to 0$ is a proper morphism.

- Blowing up at $x_\sigma$: take a basis $\ts{v_i}$, set $v_0\da \sum v_i$, and replace $\sigma$ by all subsets of $\ts{v_0,v_1,\cdots, v_n}$ not containing $\ts{v_1, \cdots, v_n}$.
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