# Wednesday, August 16

## 1.0 Background: Affine Toric Varieties

:::{.remark}
Let $S = \CC[x_1, \cdots, x_n]$.
\[
\mathbf{V}(I)=\left\{p \in \mathbb{C}^n \mid f(p)=0 \text { for all } f \in I\right\} \\
\mathbf{I}(V)=\{f \in S \mid f(p)=0 \text { for all } p \in V\} \\
\mathbb{C}[V]=S / \mathbf{I}(V)
.\]

Some facts: 

- $\CC[V] \in \IntDomain \iff I(V) \in \spec(S) \iff V$ is irreducible.
- $\Aff\Var\slice\CC (V_1\mapsvia{\phi} V_2) \mapstofrom \Alg\slice\CC(\CC[V_2]\mapsvia{\phi^*} \CC[V_1])$.
- Isomorphisms of varieties are isomorphisms of $\CC\dash$algebras.
- All maximal ideals of $\CC[V]$ are of the form $\mfm_p \da \ts{f\in \CC[V] \st f(p) = 0} \subseteq \CC[V]$.
- $R\in \Alg\slice\CC$ is of the form $R = \CC[V]$ for $V$ an affine variety iff $R\in \Alg\slice\CC^\fg$ is reduced (no nonzero nilpotents).
- *Some* open subsets of varieties are again varieties, e.g. $V_f \da \ts{p\in V\st f(p) \neq 0}$ is affine.
    - $V_f \cong \spec \CC[V]_f$, the localization at $f$, i.e. $\CC[V]_f \da \ts{{g\over f^k} \in \CC(V) \st g\in \CC(V), k\geq 0}$ where $\CC(V) \da ?$.
- $(\cstar)^n \da \CC^n\sm (\prod x_i) \subseteq \CC^n$ which has coordinate ring given by the localization $\CC[x_1,\cdots, x_n]_{\prod x_i} \cong \CC[x_1^\pm, \cdots, x_n^{\pm }]$ (Laurent polynomials).
- $V$ is normal iff $\CC[V]$ is a normal ring, i.e. integrally closed in $\CC(V)$ (all elements of $\CC(V)$ integral over $\CC[V]$, i.e. those which are roots of polynomials in $\CC[V][x]$, actually like in $\CC[V]$).
    - UFD $\implies$ normal.
- The normalization is given by $V^\nu \da \spec (\CC[V])^\nu$ where $\CC[V]^\nu \da \ts{f\in \CC(V) \st f\text{ is integral over }\CC[V]}$ (i.e. the integral closure).
- $\OO_{V, p} \da \ts{f/g\in \CC(V) \st f,g\in \CC[V], g(p) \neq 0} \contains \mfm_{V, p} \da \ts{f\in \OO_{V, p} \st f(p) = 0}$, and uniqueness makes this a local ring.
- $\T_p\dual V \da \mfm_{V, p}/\mfm_{V, p}^2$ and $T_p V \da \mods{\CC}(\T_p\dual V \to \CC)$.
- $d_p f \da \sum_{i=1}^n \dd{f}{x_i}(p) x_i$ and $T_p V \cong V(d_p(f_1), \cdots, d_p(f_\ell))$ where $I(V) = \gens{f_1,\cdots, f_\ell}$, thus $\dim \T_p V \leq n$.
- $V$ is smooth if $\dim_p \T_p V = \dim_p V$ where the latter is the maximal dimension of all irreducible components containing $p$.
- If $p$ is contained in two or more distinct irreducible components of $V$, it is singular.
- For $\dim V = d$ with $V = \gens{f_1,\cdots, f_s} \subseteq \CC^n$, a point $p$ is singular iff $\rank \Jac(f_1,\cdots, f_s)(p) = n-d$.
- $p\in V$ is smooth iff $\OO_{V, p}$ is regular local.
:::

:::{.exercise title="?"}

- Show $\CC^n$ is normal.
- Show that if $V$ is an affine variety then $V_f$ is an affine variety for any $f\in \CC[V]\smz$.
- Show that $V(x^3-y^2) \subseteq \CC^2$ is not normal.
- Show $\dim \T_p \CC^n = n$ for all $p$.
- Show $\CC^n$ is smooth.
- Find $V(x^3 - y^2)^\sing$.
- Show that a smooth irreducible affine variety is normal but the converse need not hold.
- Show that if $V_i$ are affine varieties then $\prod V_i$ is an affine variety.
- Produce a ring isomorphism which is not a $\CC\dash$algebra isomorphism.
- Describe the coordinate ring of $V\times \CC^n$ in terms of $\CC[V]$.
    Do the same for $(\cstar)^n$.
- Show that the Zariski topology on $V_1\times V_2$ is not the product topology. 
    > Hint: use $\CC^2$, and geometrically describe a basis for the product topology.
    Contradict that $V(f)$ is always closed.

:::

## 1.1: Introduction to Affine Toric Varieties

:::{.remark}

- Characters of a torus $T$: $\chi \in \Grp(T, \cstar)$.
    - Any $\vector m \in \ZZ^n$ yields a character $\chi^{\vector m} \in \Grp( (\cstar)^n, \cstar)$ where $(t_1,\cdots, t_n)\mapsto \prod t_i^{m_i}$, and all characters of $(\cstar)^n$ arise this way.
- If $\phi \in \Grp(T_1, T_2)$ with $T_i$ tori, then $\im \phi \subseteq T_2$ is a closed torus.
- If $H \subseteq T$ is an irreducible subvariety which is also a subgroup, it is also a torus.
- $M \da$ all characters of $T$.
    Think of these as monomials.
- If $T\actson W$ with $W\in \Vect\slice \CC$, there is a weight space decomposition $W = \bigoplus _{m\in M} W_m$ where $W_m \da \ts{w\in W \st t.w = \chi^m(t)w \, \forall t\in T}$.
- One-parameter subgroups/cocharacters of $T$: $\lambda \in \Grp(\cstar, T)$.
    - E.g. $\vector u\in \ZZ^n$ yields
\[
\lambda^u: \cstar &\to (\cstar)^n \\
\lambda^u(t) &\mapsto (t^{b_1}, \cdots, t^{b_n})
.\]

- $N\da$ all cocharacters of $T$.
    Think of these as monomial curves.
- Describing the bilinear pairing $M\times N\to \ZZ$: given $\chi^m$ and $\lambda^u$, get $\chi^m \circ \lambda^u: \cstar\to \cstar$ a character of $\cstar$, which are all of the form $t\mapsto t^\ell$, so set $\inp{m}{u} \da \ell$.
    - This yields $N\cong M\dual \da \zmod(M\to \ZZ)$ and $M\cong N\dual \da \zmod(N\to \ZZ)$.
    - Yields $N\tensor_\ZZ \cstar \cong T$ via $u\tensor t \mapsto \lambda^u(t)$.
- Affine toric variety: any $V\contains T_N \cong (\cstar)^n$ as a Zariski-open such that $T_N\selfmap$ extends to $T_N\actson V$ algebraically.
- The rational normal cones of degree $d$: the image of 
\[
\phi: \CC^2 &\to \CC^{d+1} \\
(s, t) &\mapsto (s^d, s^{d-1}t, \cdots, st^{d-1}, t^d)
,\]
or equivalently $V(I)$ where $I$ is generated by all $2\times 2$ minors of the following matrix: 
\[
\left(\begin{array}{ccccc}
x_0 & x_1 & \cdots & x_{d-2} & x_{d-1} \\
x_1 & x_2 & \cdots & x_{d-1} & x_d
\end{array}\right)
.\]

- Any finite set $A \subseteq M$ of size $s$ yields a toric variety $Y_A$ defined as the Zariski closure of 
\[
\Phi_A: T_N &\to \CC^s \\
t &\mapsto (\chi^{m_1}(t), \cdots, \chi^{m_s}(t))
.\]
    - Moreover $Y_A$ is an affine toric variety with character lattice $\ZZ A$ and $\dim Y_A = \rank_\ZZ \ZZ A$.
    One can compute this rank by placing the vectors of $A$ in the columns of a matrix and computing the rank of that matrix.
- Every affine toric variety is of the form $Y_A$ for $A$ a finite subset of some lattice.
- Viewing $\Phi_A: T_N \to (\cstar)^n$ as a map of tori and thus induces a morphism of lattices $\tilde \Phi_A: \ZZ^s \to M$ where $Y_A \subseteq \CC^s$ and $Y = \ts{m_1, \cdots, m_s}$.
- To describe the ideal $I(Y_A)$: let $L = \ker(\ZZ^s \to M)$, then 
\[
\left.\mathbf{I}\left(Y_{\mathscr{A}}\right)=\left\langle x^{\ell_{+}}-x^{\ell_{-}} \mid \ell \in L\right\rangle=\left\langle x^\alpha-x^\beta\right| \alpha, \beta \in \mathbb{N}^s \text { and } \alpha-\beta \in L\right\rangle 
.\]

- An ideal is toric iff it is prime and generated by binomials.
    - Here a toric ideal is a prime lattice ideal, where a lattice ideal is an ideal of the form $I_L = \gens{x^a- x^b \st a,b\in \NN^s,\, a-b\in L}$.
- For $S$ a finitely-generated monoid, regard \( \mathbb{C}[\mathrm{S}]=\left\{\sum_{m \in \mathrm{S}} c_m \chi^m \mid c_m \in \mathbb{C} \text { and } c_m=0 \text { for all but finitely many } m\right\} \) with multiplication induced by $\chi^m \chi^{m'} = \chi^{m+m'}$.
    In particular, if $S = \NN A$ for $A = \ts{m_1,\cdots, m_s}$ then $\CC[S] = \CC[\chi^{m_1},\cdots, \chi^{m_s}]$.
- $\CC[\NN^n] = \CC[x_1,\cdots, x_n]$ where $x_i = \chi^{e_i}$ for $e_i$ the standard basis of $\ZZ^n$.
    Similarly, $\CC[M] = T_N$.


:::

:::{.exercise title="?"}
- What is the torus $T$ for $C = V(x^3-y^2) \subseteq \CC^2$?
    Is this a normal variety?
- What is the torus $T$ for $V(xy-zw) \subseteq \CC^4$?
    Is this normal?

:::