# Toric Varieties

## Summaries

### Quick Criteria

:::{.remark}
Quick criteria:

- **Normal** $\iff$ **Saturated**: 
  For affines, $X = \spec \CC[S]$ where $S \subseteq M$ is a **saturated** semigroup.
  This is true for $S = S_\sigma = \sigma\dual \intersect M$ where $\sigma$ is any SCRPC. 
- **Complete/proper** $\iff$ **Full support**: 
  $X_\Sigma$ is complete iff $\supp \Sigma = N_\RR$.
- **Smooth** $\iff$ **Lattice basis**: 
  - For a **cone** $\sigma = \Cone(S)$ is smooth iff $\det S = \pm 1$, the volume of the standard lattice $\ZZ^n$. 
	- Consequences of smoothness:
		- $\CDiv(X) = \Div(X)$
		- $\Cl(X) = \Pic(X)$
  - Smooth implies simplicial, so non-simplicial cones are singular. 
  - For $p_\sigma$ the $T\dash$fixed point corresponding to $\sigma$, $T_p X \cong H$ where $H$ is a Hilbert basis for $S_\sigma$.
- **Simplicial** $\iff$ **Euclidean basis**: 
  For $\sigma = \Cone(S)$, $\sigma$ is simplicial iff $\det(S) \neq 0$.
- **Orbifold singularities** $\iff$ **Simplicial**: 
  $X_\Sigma$ has at worst finite quotient singularities iff $\Sigma$ is simplicial.
- **Projectivity** $\iff$ **Admits a strictly upper convex support function**: 
  For $h$ a support function and $D_h$ its associated divisor, the linear system $\abs{D_h}$ defines an embedding $X(\Delta) \embeds \PP^N$ iff $h$ is strictly upper convex.
  - Alternatively, $X_\Sigma$ is projective iff $\Sigma$ arises as the normal fan of a polytope.
- **Globally generated/basepoint free** $\iff$ **Upper convex support function**: 
  $\OO(D)$ is globally generated iff $\psi_D$ is upper convex.
- **Ample $\iff$ Strictly upper convex support function**:  
  $D\in \CDiv_T(X)$ is ample iff $\psi_D$ is strictly upper convex.
- **Very ample $\iff$ ample and semigroup generation**: for $\Sigma$ complete, $D$ is very ample iff $\psi_D$ is strictly upper convex **and** $S_\sigma$ is generated by $\ts{u-u(\sigma) \st u\in P_D \intersect M}$, or equivalently the semigroup $\ts{u-u' \st u'\in P \intersect M}$ is saturated in $M$.
  - For $\PP^n$: $D = \sum a_i D_i$ is globally generated iff $\sum a_i \geq 0$ and ample $\iff \sum a_i > 0$.
  - For $\FF_m$: $D = \sum a_i D_i$ is globally generated iff $a_2 + a_4 \geq 0,\, a_1 + a_3 \geq m a_1$, $\Pic(\FF_n) = \gens{D_1, D_4}$, and $D = aD_1 + bD_4$ is ample iff $a,b > 0$.
  - For $\dim X_\Sigma = 2$ and $X$ complete: ample $\iff$ very ample.

- **$\QQ\dash$factorial $\iff$ simplicial**: 
  iff every cone is simplicial.

- **Fundamental groups**:
  - For $U_\sigma$ affine, $U_\sigma \cong \AA^k \times \GG_m^{n-k}$ so $\pi_1 U_\sigma \cong \ZZ^{n-k}$ since $\GG_m^{n-k}\homotopic (S^1)^{n-k}$.
  - Can write $\pi_1 U_\sigma = N/N_\sigma$ where $N_\sigma$ is the sublattice generated by $\sigma$.
  - By a Van Kampen argument, $\pi_1 X_\Sigma = N/N'$ where $N' = \gens{\sigma \intersect N \st \sigma \in \Sigma}$:
  \[
  \pi_1 X_\Sigma = \pi_1 \union U_{\sigma} = \colim \pi_1 U_\sigma = \colim N/N_\sigma = N / \Sum N_\sigma = N/N'
  .\]

- **Euler characteristic**: $\chi X_\Sigma = \size \Sigma(n)$.
  - Why: $H^i(U_\sigma; \ZZ) = \Extpower^i M(\sigma)$ where $M(\sigma ) \da \sigma\dual \intersect M$, so one gets a spectral sequence 
  \[
  E_1^{p, q} = \bigoplus _{I^p = i_0< \cdots < i_p} H^q(U_{\sigma_{I^p}}; \ZZ) \abuts H^{p+q}(X_\Sigma; \ZZ), \qquad \sigma_{I^p} = \sigma_{i_0} \intersect \cdots \sigma_{i_p}, \sigma_{i_j}\in \Sigma(n) \\ \\
  \leadsto E_1^{p, q} = \bigoplus _{I^p} \Extpower^q M(\sigma_{I^p}) \abuts H^{p+q}(X_\Sigma; \ZZ) \\\\
  \implies \chi X_\Sigma = \sum (-1)^{p+q} \rank_\ZZ E_1^{p, q} = \size \Sigma(n)
  ,\]
  using that
  \[
\sum (-1)^{q} \rank_\ZZ \Extpower^q M(\tau) = 
\begin{cases}
0 & \dim \tau < n 
\\
1 & \dim \tau = n.
\end{cases}
  .\]

- **Higher homology**:
  - If all maximal cones of $\Sigma$ are $n\dash$dimensional, $H^2(X_\Sigma; \ZZ) \cong \Pic(X_\Sigma)$.

- **Global sections**: for $D\in \Div_T(X)$, $P_D$ its associated polyhedron,
\[
H^0(X; \OO_X(D)) = \bigoplus _{m\in P_D \intersect M} \CC\, \chi^m
.\]

- **Betti numbers**:
\[
\beta_{2k} = \sum_{i=k}^n (-1)^{i-k} {i\choose k} \size\Sigma(n-i)
.\]

- **Canonical bundles/divisors**: $\omega_{X_\Sigma} \da \det \Omega_{X_\Sigma/k} = \OO(K_{X_\Sigma})$ where $K_{X_\Sigma} = -\sum_{\rho_i} D_i$.
  - For a smooth complete surface with $D_i^2 = -d_i$, 
  \[
  K^2 = \sum D_i^2 + 2d = -\sum d_i + 2d = -(3d-12) + 2d = 12-d
  .\]

- **Degree = $n! \cdot \vol(P)$** (for $X_P$ projective)
:::

:::{.remark}
Some common counterexamples:

- An ample divisor that is not very ample: $P \da \mathrm{Conv}(\tv{0,0,0}, \tv{0,1,1}, \tv{1,0,1}, \tv{1,1,0})$; then take $D_P$. $X_P$ is a double cover of $\PP^3$ branched along the 4 boundary divisors.
:::


### Cones and Lattices

:::{.remark}
\envlist

- **Characters**: for groups $G$, a map $\chi\in \Grp(G, \cstar)$.
  For $G= T = (\cstar)^n$, there is an isomorphism
  \[
  \ZZ^n &\iso \Grp(T, \cstar) \\
  m = \tv{m_1,\cdots, m_n} &\mapsto \chi_m: \tv{t_1,\cdots, t_n} \mapsto \prod t_i^{m_i}
  .\]
  Generally set $M \da \Grp(T, \cstar)$, the character lattice.
  - $M$ is a lattice, $M_\RR \da M\tensor_\ZZ \RR$ is its associated Euclidean space.

- **Cocharacters / one-parameter subgroups**: for groups $G$, a map $\lambda \in \Grp(\cstar, G)$.
  For $G = T = \cstar$, there is again an isomorphism
  \[
  \ZZ^n &\mapsto \Grp(\cstar, T) \\
  u =\tv{u_1,\cdots, u_n} &\mapsto \lambda^u: t\mapsto \tv{t^{u_1}, \cdots, t^{u_n}}
  .\]
  Define $N \da \Grp(\cstar, T)$ the cocharacter lattice.

  - $N$ is a lattice, $N_\RR \da N\tensor_\ZZ \RR$ its associated euclidean space.


- There is a perfect pairing
\[
\inp\wait\wait: M\times N &\to \ZZ \\
,\]
defined using the fact that if $m\in M, n\in N$ then $\chi^m \circ \lambda^n \in \Grp(\cstar, \cstar)$ is of the form $t\mapsto t^\ell$, so set $\inp{m}{n} \da \ell$.

  - Thus $M = \Grp(M, \ZZ)$ and $N = \Grp(N, \ZZ)$.
  - How to recover the torus:
  \[
  N \tensor_\ZZ \cstar &\to T \\
  u\tensor t &\mapsto \lambda^u(t)
  .\]

- $\Delta$ is a **fan**, a collection of **strongly convex rational polyhedral cones**:
  - **Cone**: $0\in \sigma$ and $\RR_{\geq 0} \sigma \subseteq \sigma$.
  - **Strongly convex**: contains no nonzero subspace, i.e. no line through $\vector 0 \in N_\RR$.
    Equivalently, $\dim \sigma\dual = n$.
  - **Rational**: generated by $\ts{v_i} \subseteq N$, i.e. of the form $\Cone(S)$ for $S \subseteq N$.
- **Dual cones**:
\[
\sigma\dual &\da \ts{ u\in M \st \inp u v \geq 0 \,\,\forall v\in M_\RR }
.\]
  - If $\sigma\dual = \Intersect_{i=1}^s H_{m_i}^+$ for $m_i \subseteq \sigma\dual$ then $\sigma\dual = \Cone(m_1,\cdots, m_s)$.
- **Hyperplanes** and **closed half-spaces**:
\[
H_m &\da \ts{u\in N_\RR \st \inp m u = 0} \subseteq N_\RR \\
H_m^+ &\da \ts{u\in N_\RR \st \inp m u \geq 0} \subseteq N_\RR 
.\]

- **Face**: $\tau \leq \sigma$ is a face iff $\tau$ is of the form $\tau = H_m \intersect \sigma$ for some $m\in \sigma\dual \subseteq M_\RR$.
- **Facet**: codimension one faces, $\Sigma(n-1)$ where $n\da \dim N$.
- **Ray**: dimension 1 faces, $\Sigma(1)$.

- The **semigroup** of a cone:
\[
S_\sigma &\da \sigma\dual \intersect M = \ts{ u\in M \st \inp u v \geq 0 \,\,\forall v\in \sigma }
.\]
- The **semigroup algebra** of a semigroup:
\[
\CC[S] \da \ts{\sum_{s\in S} c_s \chi^s \st c_s \in \CC, c_s = 0 \ae}, \qquad \chi^{m_1}\cdot \chi^{m_2} \da \chi^{m_1 + m_2}
.\]

- **Simplicial**: the generators can be extended to an $\RR\dash$basis of $N_\RR$.
  E.g. not simplicial:

![](figures/2022-10-19_18-23-05.png)

- **Smooth**: the minimal generators can be extended to a $\ZZ\dash$basis of $N$.
  - Checking $T_p X$: $m$ is **decomposable** in $S_ \sigma$ iff $m = m_1 + m_2$ with $m_i\in S_{ \sigma}$; the maximal ideal at $p$ corresponding to $\sigma$ is $\mfm_p = \ts{\chi^m \st m\in S_ \sigma}$, and $\mfm_p/\mfm_p^2 = \ts{\chi^m \st m \text{ is indecomposable in } S_ \sigma}$.
  This exactly corresponds to a Hilbert basis.
- **Facet**: face of codimension 1.
- **Edge**: face of dimension 1.
  Note that facets = edges in $\dim N = 2$.

- **Saturated**: $S$ is saturated if for all $k\in \NN\smz$ and all $m\in M$, $km\in S \implies m\in S$.
  Any SCRPC is saturated.
  - E.g. $S = \ts{(4,0), (3,1), (1,3), (0, 4)}$ is not saturated since $2\cdot(2,2) = (4, 4) \in \NN S$ but $(2,2)\not\in S$.
- **Normalization**: in the affine case, write $X = \spec \CC[S]$ with torus character lattice $M = \ZZ S$, take a finite generating set $S'$, and set $\sigma = \Cone(S')\dual$.
  Then $\spec \CC[\sigma\dual \intersect M]\to X$ is the normalization.

- **Distinguished points**: each strongly convex $\sigma \leadsto \gamma_\sigma \in U_\sigma$ a unique point corresponding to the semigroup morphism $m\mapsto \indic(m\in \sigma\dual \intersect M)$, which is $T\dash$fixed iff $\sigma$ is full-dimensional.
- **Orbits**: $\Orb( \sigma) = T. \gamma_\sigma$, and $V(\sigma)\da \cl \Orb( \sigma)$.
- **Orbit-Cone correspondence**: there is a correspondence 
\[
\ts{\text{Cones } \sigma \in \Sigma} &\mapstofrom \ts{T\dash\text{orbits in } X_\Sigma} \\
\sigma &\mapsto \Orb(\sigma) \da T.\gamma_{\sigma} = \ts{\gamma: S_\sigma \to \CC\st \gamma(m) \neq 0 \iff m\in \sigma\dual \intersect M} \cong \Grp(\sigma \intersect M, \cstar)
,\]
  where $\dim \Orb( \sigma) = \codim_{N_\RR} \sigma$, and $\tau \leq \sigma \implies \cl \Orb(\tau) \contains \cl\Orb( \sigma)$ and in fact $\cl \Orb(\sigma) = \disjoint_{\tau\leq \sigma} \cl \Orb( \tau)$.
- **Star**: define $N_\tau \da \ZZ \gens{\tau \intersect N}$ and $N(\tau)\RR \da N_\RR / (N_\tau)_\RR$ and $\bar\sigma$ for the image of $\sigma$ under the quotient map, then
\[
\mathrm{Star}(\tau) \da \ts{\bar \sigma \subseteq N(\tau)_\RR \st \sigma\leq \tau } \subseteq N(\tau)_\RR
.\]
  This is always a fan, and $V(\tau) = X_{\mathrm{Star}(\tau)}$.
- **Star subdivision**: for $\sigma = \Cone(S)$ for $S \da \ts{u_1,\cdots, u_n}$, set $u_0 \da \sum u_i$ and take $\Sigma'(\sigma)$ defined as the cones generated by subsets of $\ts{u_0, u_1, \cdots, u_n}$ not containing $S$.
  The star subdivision of $\Sigma$ along $\sigma$ is $\Sigma^\star(\sigma) \da (\Sigma \smts{\sigma}) \union \Sigma'( \sigma)$.
- **Blowups**: $\phi: X_{\Sigma^\star(\sigma)}\to X_{\Sigma}$ is the blowup at $\gamma_ \sigma$.


:::

### Divisors

:::{.remark}
\envlist

- **(Weil) divisor**: $\Div(X) = \ts{\sum n_i V_i \st V_i \subseteq X, \codim V_i = 1}$.
	- $\OO_X(D)$: the (coherent) sheaf associated to a Weil divisor $D$.
- **Cartier divisor**: $\CDiv(X) = H^0(X; \mck_X\units/\OO_X\units)$, the quotient of rational functions by regular functions.
  For $X$ normal, equivalently locally principal (Weil) divisors, so $D \leadsto \ts{(U_i, f_i)}$  where $\ro{D}{U_i} = \div(f_i)$.
  - **$\QQ\dash$Cartier divisor**: A $\QQ\dash$divisor $D =\sum n_i D_i$ with $n_i\in \QQ$ is $\QQ\dash$Cartier when $mD$ is Cartier for some $m\in \ZZ_{\geq 0}$.
  - **$\QQ\dash$factorial**: every prime divisor is $\QQ\dash$Cartier.
- **Ray divisors**: every $\rho\in \Sigma(1)$ defines a divisor $D_\rho \da V(\rho) \da \cl\Orb( \rho)$.
- **Very Ample**: $\mcl$ which defines a morphism into $\PP H^0(X; \mcl) \cong \PP^N$.
- **Ample**: $\mcl$ is basepoint free and some power $\mcl^n$ is very ample.
  - $D$ is (very) ample iff $\OO_X(D)$ is (very) ample, i.e. $D$ is ample iff $nD$ is very ample for some $n$.
- **Upper convex**: $f(n_1 + n_2) \leq f(n_1) + f(n_2)$.
  - **Strictly upper convex**: $\sigma_1\neq \sigma_2 \implies f_{\sigma_1} \neq f_{\sigma_2}$.
- **Linearly equivalent divisors**: $D_1\sim D_2 \iff D_1 - D_2 = \div(f)$ for some $f$.
- **Complete linear systems**: $\abs{D} = \ts{D'\in \Div(X) \st D'\sim D}$.
- **Support function**: $\phi: \supp \Sigma \to \RR$ where $\ro{\phi}{\sigma}$ is linear for each cone $\sigma$.
  - **Integral** with respect to $N$ iff $\phi(\supp \Sigma \intersect N) \subseteq \ZZ$. 
  Defines a set of integral support functions $\SF(\Sigma, N)$.

- The class group complement exact sequence: for $D_1,\cdots, D_n \in \Div(X)$ distinct,
\[
\ZZ^n &\to \Cl(X) \surjects \Cl(X\sm\union D_i) \\
e_1 &\mapsto [D_i]
.\]
- $\OO_X(D)$ is the sheaf 
\[
U\mapsto \ts{f\in \mck(X)\units(U) \st \div(f) + \ro{D}{U} \geq 0 \in \Cl(U) }
.\]
  Then $D\in \CDiv(X) \iff \OO_X(D) \in \Pic(X)$.

- The toric class group exact sequence:
\[
M &\to \Div_T(X) \surjects \Cl(X) \\
m &\mapsto \div(\chi^m) = \sum_\rho \inp{m}{u_\rho} [D_\rho]
\]
where $u_\rho$ are minimal ray generators.

:::

### Polytopes

:::{.remark}
\envlist

- **Supporting hyperplanes**: the positive side of an affine hyperplane
\[
H_{u, b} &\da \ts{m\in M_\RR \st \inp m u = b} \\
H_{u, b}^+ &\da \ts{m\in M_\RR \st \inp m u \geq b}
.\]
  - If $P$ is full dimensional and $F\leq P$ is a facet, then $F = P \intersect H_{u_F, -a_F}$ for a unique pair $(u_F, a_F) \in N_\RR\times \RR$.

- **Polytope**: the convex hull of a finite set $S \subseteq N_\RR$ or an intersection of half-spaces:
\[
P = \ts{\sum_{v\in S} \lambda_v v \st \sum \lambda_v = 1} = \Intersect _{i=1}^s H_{u_i, b_i}^+ 
.\]
- **Simplex** $\dim P = d$ and there are exactly $d+1$ vertices.
- **Simple**: $\dim P = d$ and every vertex is the intersection of exactly $d$ facets.
- **Simplicial**: all facets are simplices.
  - E.g. simple but not simplicial: the cube in $\RR^3$, since each vertex meets 3 edges but a square is not a simplex. 
  -E.g. Simplicial but not simple: the octahedron in $\RR^3$, since each vertex meets 4 edges but each face is a triangle.
- **Combinatorial equivalence**: $P_1\sim P_2$ iff there is a bijection $P_1\to P_2$ preserving intersections, inclusions, and dimensions of all faces.

- **Polar dual**: for $P \subseteq M_\RR$,
\[
P^\circ = \ts{u\in N_\RR \st \inp m u \geq - 1\,\, \forall m\in P}
.\]
  - Trick: for $P \subseteq M_\RR$ with $0\in P$,
  \[
  P = \ts{m\in M_\RR \st \inp m {u_F} \geq -a_F,\, F \in \mathrm{Facets}(P) } \\
  \implies P^\circ = \mathrm{Conv}(\ts{ a_F\inv u_F }) \subseteq N_\RR
  .\]
  E.g. write the square as $\ts{\inp{m}{\pm e_i}\geq -1}$, then $a_F = 1$ for all $F$:
  ![](figures/2022-10-19_18-35-59.png)

- **Cone on a polytope**: $C(P) \da \Cone(P\times \ts{1}) \subseteq M_\RR \times \RR$, the set of cones through all proper faces of $P$.
- **Normal**: $\qty{kP \intersect M} + \qty{\ell P \intersect M} \subseteq (k+\ell)P \intersect M$, or equivalently $k\cdot (P \intersect M) = (kP) \intersect M$, or equivalently $(P \intersect M)\times\ts{1}$ generates $C(P) \intersect (M\times \ZZ)$ as a semigroup.
  - If $P \subseteq M_\RR$ is a full-dimensional lattice polytope with $\dim P \geq 2$, then $kP$ is normal for all $k\geq \dim P - 1$.
  - Normal implies very ample.
  - $P\leadsto \mcl_P \in \Pic(X_P)$ 
  - $P \intersect M \leadsto H^0(X_P; \mcl_P)$.

- **Reflexive**: a polytope $P$ with facet presentation 
\[
P = \ts{m\in M_\RR \st \inp{m}{\mu_F} \geq -1 \forall F\in \mathrm{Facets}(P)}
.\]
  Implies that $\int(P) \intersect M = \ts{\vector 0}$, and $P^\circ = \mathrm{Conv}(\ts{u_F \st F\in \mathrm{Facets}(P)})$.
- **Polyhedron of a divisor $P_D$**: write $D = \sum_{\rho} a_{\rho} D_{\rho}$, for any $m\in M$, $\div(\chi^m) + D \geq 0 \implies \inp{m}{\rho} \geq a_{\rho} \implies \inp{m}{\rho} \geq - a_\rho$, so set
\[
P_D \da \ts{ m\in M_\RR \st \inp m \rho \geq a_\rho \, \forall \rho \in \Sigma(1)}
.\]

- **Divisor of a polytope**: $D_P = \sum_F a_F D_F$ where $P = \ts{m \st \inp m{u_F} \geq -a_F}$.
  - $D_P$ is always the pullback of $\OO_{\PP^N}(1)$ along the embedding.
- **Very ample polytopes**: for every vertex $v$, the semigroup $\ts{m' - v \st m'\in P \intersect M}$ is saturated in $M$.
  - Gives an embedding $X \embeds \PP^N$ where $N = \size(P \intersect M) - 1$.
- The **toric variety of a polytope**: if $P \intersect M = \ts{m_1,\cdots, m_s}$ and $P$ is full dimensional very ample, then writing $T_N$ for the torus of $N$, 
\[
X_{P} \da \cl \im \phi,\qquad \phi: T_N &\to \PP^{s-1} \\
t &\mapsto \tv{\chi^{m_1}(t) : \cdots : \chi^{m_s}(t)}
.\] 

  - Vertices $m_i$ correspond to $U_{\sigma_i}$ for $\sigma_i = \Cone(P \intersect M - m_i)\dual$:

  ![](figures/2022-10-19_19-08-06.png)

- **Smooth**: $P$ is smooth iff for all vertices $v\in P$, $\ts{w_E - v\st E\text{ is an edge containing }v}$ can be extended to a $\ZZ\dash$basis of $M$, where $w_E$ is the first lattice point on $E$.

:::

### Singularities and Classification

:::{.remark}
\envlist

- **Gorenstein**: $X$ normal where $K_X \in \CDiv(X)$ is Cartier.
- **Normal**: all local rings are integrally closed domains.
- **Complete**: proper over $k$. E.g. for varieties, just universally closed.
- **Factorial**: all local rings are UFDs.
- **Fano**: $-K_X$ is ample.
- **del Pezzo**: a smooth Fano surface.

:::

:::{.remark}
Classification of smooth complete toric varieties:

- $\dim \Sigma = 2, \size \Sigma(1) = 3$: without loss of generality $\rho_1 = e_1, \rho_2 = e_2$.
  Then $\rho_3 = a e_1 + be_2$ with $a,b< 0$ to ensure $\supp \Sigma = \RR^2$, and determinants for $\abs{a} = \abs{b} = 1$, so $(-1, 1)$.
- $\dim \Sigma = 2, \size \Sigma(1) = 4$: without loss of generality $\rho_1 = e_1, \rho_2 = e_2$.
  Then determinant conditions for $\rho_3 = (-1, b)$ and $\rho_4 = (a, -1)$, and $\det\matt{-1}{a}{b}{-1} = 1-ab = \pm 1 \implies ab=0,2$, so $(a,b) = (2,1), (1,2), (-2, -1), (-1,-2)$.
- $\dim \Sigma = 2, \size\Sigma(1) = d$, smooth: $\Bl_{p_1,\cdots, p_\ell} X$ for $X = \PP^2$ or $\FF_a$ for some $a$ and $p_i$ torus fixed points.
:::

### Examples

:::{.question}
Things you can figure out for every example:

- Given $\Delta$, for $\sigma\in \Delta$, 
  - What is $\sigma\dual$? 
  - Generators for $S_\sigma$? 
  - Describe $U_\sigma$ and $X(\Delta)$.
  - What are the transition functions for $U_{\sigma_1} \to U_{\sigma_2}$ when $\sigma_1 \intersect \sigma_2 = \tau$ intersect in a common face?
- What are the $T\dash$invariant points? 
  - What are the $T\dash$invariant divisors $D_{\rho_i}$? 
  - What are all of the $T\dash$orbit closures of various dimensions?
- Is $X(\Delta)$ smooth?
  - Which cones $\sigma\in \Delta$ are smooth? 
  - What is the canonical resolution of singularities? 
  - What is the tangent space at each $T\dash$invariant point?
- What is the associated polytope $P_\Delta$? What is its polar dual $P_\Delta^\circ$?
- What are the intersection numbers $D_{\rho_i} \cdot D_{\rho_j}$?
  - What are the self-intersection numbers $D_{\rho_i}^2$?
- What is $\Div_T(X)$? $\CDiv_T(X)$?
  - Which divisors are ample? Very ample? Globally generated?
- What is $\Cl(X)$? $\Pic(X)$?
- What is $K_X$?
  - Is $K_X$ ample?
- Is $X(\Delta)$ projective?
- What is $H^0(X(\Delta); \OO(D) )$ for $D\in \Div_T(X)$?
- What is the Poincaré polynomial of $X(\Delta)$? (I.e. what are the Betti numbers?)


:::

:::{.example title="of varieties"}
Some useful explicit varieties:

- $V(x^3-y^2)$ with torus $T = \ts{\tv{t^2, t^3} \st t\in \cstar}$.
- $V(xy-zw)$ with torus $T = \ts{\tv{a,b,c,abc\inv} \st a,b,c,d\in \cstar}$.
- $V(xz-y^2)$, note $V(x, y)\in \Div(X) \sm \CDiv(X)$.
- $\im([x:y] \mapsto [x^3: x^2y : xy^2 : y^3])$ the twisted cubic.
  Corresponds to $\sigma\dual = \ts{(3,0), (2,1), (1,2), (0, 3)}$.
- The **rational normal scroll**: $V\qty{2\times 2\text{ minors of } \left[\begin{array}{lll}
x_0 & x_1 & y_0 \\
x_1 & x_2 & y_1
\end{array}\right]}$ is the image of $\tv{s,t} \mapsto \tv{1:s:s^2:t:st}$.
- The Segre variety: $\spec \CC[x_1y_1, x_1 y_2, \cdots, x_1 y_n, x_2 y_1, \cdots, x_m y_1, \cdots x_m y_n]$.

:::

:::{.example title="of fans"}
\envlist

- $(\CC\units)^n$: Take $\Delta = \ts{ \sigma_0 = \NN\gens{0}} \subseteq N$ with $\dim N = n$ yields $S_{\sigma_0} = \NN\gens{\pm e_1\dual,\cdots, \pm e_n\dual} = M$ for so $X(\Delta) = \spec \CC[x_1^{\pm 1},\cdots, x_n^{\pm 1}] = (\GG_m)^n$.
- $\CC^n$: Take $\Delta = \Cone(\sigma_0 = \NN\gens{ e_1,\cdots, e_n} )$ yields the positive orthant $S_{\sigma_0} = \NN\gens{e_1\dual,\cdots, e_n\dual} \subseteq M$, so $X(\Delta) = \spec \CC[x_1,\cdots, x_n] = \AA^n$.
- The quadric cone: $\Delta = \Cone(\sigma_1 = \NN\gens{e_2, 2e_1 - e_2})$ yields $S_{\sigma_1} = \NN\gens{e_1\dual, e_1\dual + e_2\dual, e_1\dual + 2e_2\dual}$ so $X(\Delta) = \spec \CC[x, xy, xy^2] = \spec \CC[u,v,w]/(v^2-uw)$:

![](figures/2022-10-18_15-33-37.png)

![](figures/2022-10-18_15-33-47.png)

- $\PP^1$: Take $\Delta = \ts{\RR_{\geq 0}, 0, \RR_{\leq 0}}$ and glue along overlaps to get $X(\Delta) = \PP^1$ with gluing maps $x\mapsto x\inv$:

![](figures/2022-10-18_15-36-53.png)

- $\Bl_1 \CC^2$: Take $\sigma_0 = \NN\gens{ e_2, e_1+e_2}$ and $\sigma_1 = \NN\gens{e_1+e_2, e_1}$ to get $U_{ \sigma_0} = \spec \CC[x, x\inv y]$ and $U_{ \sigma_1} = \spec \CC[y, xy\inv]$, both copies of $\CC^2$:

![](figures/2022-10-18_15-39-10.png)

  Why this is a blowup of $\CC^2$: write $\Bl_1 \CC^2 = V(xt_1 - yt_0) \subseteq \CC^2\times \PP^1$ for $\PP^1 = \ts{\tv{t_0: t_1}}$.
  Take the open cover $U_i = D(t_i) \cong \CC^2$, where coordinates on $U_0$ are $x, t_1/t_0 = x\inv y$ and on $U_1$ are $y, t_0/t_1 = xy\inv$ and glue.

- $\PP^2$: take $\Delta = \Cone(e_1, e_2, -e_1-e_2)$:

  ![](figures/2022-10-18_15-42-04.png)

  This has dual cone:

  ![](figures/2022-10-18_15-42-18.png)

  Each $U_{\sigma_i} \cong \CC^2$ with coordinates $(x,y), (x\inv, x\inv y), (y\inv, xy\inv)$ respectively for $U_i$. Glue to obtain $x=t_1/t_0, y=t_2/t_0$.

- $F_a$ the Hirzebruch surface: take $\Cone(e_1, -e_2, -e_1, -e_1 + ae_2)$ to get

  - $U_{\sigma_1} = \spec \CC[x,y]$,
  - $U_{\sigma_2} = \spec \CC[x,y\inv]$,
  - $U_{\sigma_3} = \spec \CC[x\inv,x^{-a} y\inv]$,
  - $U_{\sigma_4} = \spec \CC[x\inv ,x^a y]$,

  which patch in the following way:

  ![](figures/2022-10-18_15-45-17.png)

  Project to $y=0$ to get the patching $x\mapsto x\inv$, so a copy of $\PP^1$.
  Patching in the fiber direction, e.g. $U_{\sigma_1}$ and $U_{\sigma_2}$, gives a copy of $\CC \times \PP^1$.
  Thus this is a bundle $\PP^1\to \mce \to \PP^1$.


- $\CC\times \PP^1$: todo.
- $\PP^1 \times \PP^1$: todo.
- $\CC^a \times \PP^b$: todo.
- $\PP^a \times \PP^b$: todo.

:::

:::{.example title="of polytopes"}

- Hirzebruch surfaces:

![](figures/2022-12-03_20-09-23.png)

- $(\PP^2, \OO(1))$: take $P = \mathrm{Conv}(0, e_1, e_2)$, so $X_P = \cl \Phi_P$ where
\[
\Phi_P: (\cstar)^2 &\to \PP^2 \\
(s,t) &\mapsto [1: s: t]
,\]
which is the identity embedding corresponding to $\OO(1)$ on $\PP^2$.
  - $2P$ yields 
  \[
  \Phi_{2P}: (\cstar)^2 &\to \PP^5 \\
  (s,t) &\mapsto [1: s: t : s^2: st: t^2]
  ,\]
  the Veronese embedding corresponding to $\OO(2)$ on $\PP^2$.


:::

:::{.example title="Projective spaces"}
Some useful facts about $\PP^n$:

- The torus embedding is
\[
(\cstar)^n &\embeds \PP^n \\
\tv{a_1,\cdots, a_n} &\mapsto \tv{1: a_1 : \cdots : a_n}
.\]
- The torus action is 
\[
(\cstar)^n &\actson \PP^n \\
\tv{t_1,\cdots, t_n} . \tv{x_0: x_1:\cdots:x_n} &= \tv{x_0: t_1 x_1:\cdots:t_n x_n}
.\]


:::


:::{.example title="of class groups and Picard groups"}

![](figures/2022-10-20_00-07-20.png)

![](figures/2022-10-20_00-10-35.png)

:::