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2022-06-20

Reading: [@CLS11], see LitNote-Cox et al.-2011-Toric varieties-CLS11.
- See examples of toric varieties
Recorded lectures on toric varieties: https://www.youtube.com/playlist?list=PLkLlU-tx2p68VRT9w5LwBxkSxT_J6rwLb #resources/videos #resources/full-courses

Standard toric structures:

Set $\one_n \coloneqq{\left[ {1, \cdots, 1} \right]}$. Some standard toric structures on varieties:

$({\mathbf{C}}^n, ({\mathbf{C}}^{\times})^n, \one_n)$ where $\mathbf{t}.\mathbf{z} = {\left[ {t_1z_1, \cdots, t_n z_n} \right]}$ is the diagonal action.
$({\mathbf{P}}^n, ({\mathbf{C}}^{\times})^n, \one_{n+1})$ where $\mathbf{t} . \mathbf{z} = {\left[ {z_0: t_1z_1: \cdots : t_n z_n} \right]}$ is almost a diagonal action.

Then if $\phi: {\mathbb{T}}\hookrightarrow{\mathbb{T}}^n$ is any monomorphism of torii, one can define an associated toric variety $X = { \operatorname{cl}} _{{\mathrm{zar}}}(\operatorname{im}\phi)$ where $t.\mathbf{z} \coloneqq{\phi(t)}.\mathbf{z}$ in the standard structure.

Example: $(t_1, t_2) \mapsto (t_1t_2^{-1}, t_1t_2, t_1)$ yields $X = V(xy-z^2)$. Example: $(t_1, t_2, t_3)\mapsto (t_1, t_2, t_3, t_1t_2t_3^{-1})$ yields $V(xy-zw)$.

Lattices: finitely generated torsionfree (and hence free) ${\mathbf{Z}}{\hbox{-}}$modules $N$. Has an associated $N_{\mathbf{Q}}\coloneqq N\otimes_{\mathbf{Z}}{\mathbf{Q}}\in {}_{{\mathbf{Q}}}{\mathsf{Mod}} $.

Cones: make sense in ${}_{{\mathbf{Q}}}{\mathsf{Mod}} ^{\mathrm{fg}}$, any subset of the form ${\mathbf{Q}}_{\geq 0}\left\{{v_0, \cdots, v_n}\right\} \coloneqq\left\{{\sum a_i v_i {~\mathrel{\Big\vert}~}a_i\in {\mathbf{Q}}_{\geq 0}}\right\}$.

Example: ${ \mathrm{Cone} }(v_1,v_2)$ for $v_1 = {\left[ {1,1} \right]}$ and $v_2 = {\left[ {1,2} \right]}$: attachments/Pasted%20image%2020220621000455.png

Can always carve by hyperplanes using finitely many linear forms, i.e. $\sigma = \left\{{v\in V {~\mathrel{\Big\vert}~}u_1(v) \geq 0, \cdots, u_s(v) \geq 0}\right\}$ for some $u_i\in U\coloneqq\mathop{\mathrm{Hom}}_{\mathbf{Z}}(V, {\mathbf{Q}})$. Yields a dual cone $\sigma {}^{ \vee }= \left\{{u\in U {~\mathrel{\Big\vert}~}{ \left.{{u}} \right|_{{\sigma}} } \geq 0}\right\}$: attachments/Pasted%20image%2020220621000837.png Described as the intersection of half-spaces $v_1^\perp, v_2^\perp$ where the normals are chosen to point into the cone’s interior. Need linear forms that vanish along the lines for $v_i$.

Faces: cut out by a linear form coming from the dual. $\tau$ is a face of $\sigma$ iff there exists a $u\in\sigma {}^{ \vee }$ with $\tau = u^\perp \cap\sigma = \left\{{v {~\mathrel{\Big\vert}~}u(v) = 0}\right\}$. A face is pointed if it contains the origin.

Example: try to describe the faces in terms of linear forms:

attachments/Pasted%20image%2020220621001606.png

Fans in ${}_{{\mathbf{Q}}}{\mathsf{Mod}} ^{\mathrm{fg}}$: a finite set of pointed cones closed under taking faces and intersections.

Example: attachments/Pasted%20image%2020220621002019.png

One-parameter subgroups: $\lambda \in \mathop{\mathrm{Hom}}_{ \mathsf{Alg} {\mathsf{Grp}}}({{\mathbf{C}}^{\times}}, {\mathbb{T}})$. Consider $\lim_{t\to 0} \lambda(t).x_0$ where $x_0$ is a basepoint in $X$. Say that $\lambda$ converes iff it extends over the puncture to a map $\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu$ from ${\mathbf{C}}$. Define the limit as $\lim \lambda \coloneqq\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu(0)$.

Important fact: there are only finitely many possible limit points, despite infinitely many such subgroups. Idea: $t=1$ is at the center, and $t\to 0$ approaches the boundary. Can take derivative at $t=1$ to consider the finitely many directions emanating from $x_0$:

attachments/Pasted%20image%2020220621134015.png

Example: take the standard toric structure ${\mathbf{P}}^2, {\mathbb{T}}^2, {\left[ {1,1,1} \right]}$; for $\mathbf{v} \coloneqq(v_1,v_2)\in {\mathbf{Z}}^2$ there is a 1-parameter subgroup $\lambda_{\mathbf{v}}: t\mapsto {\left[ {t^{v_1}, t^{v_2}} \right]}$. Then $\lambda(t){\left[ {1,1,1} \right]} = {\left[ {1,t^{v_1}, t^{v_2}} \right]} \overset{t\to 0}\longrightarrow{\left[ {1,0,0} \right]}$ if$v_1,v_2>0$. Taking the derivative recovers $v_1,v_2$, so drawing all such $\lambda_{\mathbf{v}}$ yields the open 1st quadrant of ${\mathbf{Z}}^2$.

$\Lambda({\mathbb{T}}) \coloneqq\mathop{\mathrm{Hom}}_{ \mathsf{Alg} {\mathsf{Grp}}}({{\mathbf{C}}^{\times}}, {\mathbb{T}})$, the set of all 1-parameter subgroups, is a lattice where operations are defined pointwise.

Example: $\Lambda({\mathbb{T}}^n) { \, \xrightarrow{\sim}\, }{\mathbf{Z}}^n$ by $\mathbf{v} \in {\mathbf{Z}}^n \mapsto \lambda_{\mathbf{v}}: t \mapsto {\left[ {t^{v_1}, \cdots, t^{v_n}} \right]}$ with inverse $\lambda \mapsto {\frac{\partial \lambda}{\partial t}\,}(1)$.

Normal varieties: having at worst normal singularities.

Define $\sigma(x) = { \operatorname{cl}} _{\mathrm{zar}}{ \mathrm{Cone} }\qty{\left\{{\lambda {~\mathrel{\Big\vert}~}\lim \lambda = x}\right\}} \subseteq \Lambda_{\mathbf{Q}}({\mathbb{T}})$, then if $(X, {\mathbb{T}}, x)$ is a normal toric variety, $\Sigma(X) \coloneqq\left\{{\sigma(x) {~\mathrel{\Big\vert}~}x\text{ is a limit point}}\right\}$ is a fan in $\Lambda({\mathbb{T}})$.

If the variety is affine (not necessarily normal), then there is a unique closed torus orbit ${\mathbb{T}}.x \hookrightarrow X$ and one can write $\Sigma(X) = \left\{{\tau {~\mathrel{\Big\vert}~}\tau\leq \sigma(x)}\right\}$.

Morphisms of toric varieties: pairs $\phi: X\to X'$ and $\tilde \phi: {\mathbb{T}}\to {\mathbb{T}}'$; where it is not necessary for $\tilde \phi = { \left.{{\phi}} \right|_{{{\mathbb{T}}}} }$. Morphisms of lattice fans: ${\mathbf{Z}}{\hbox{-}}$module morphisms when send cones to cones. Example: $(v_1, v_2) \mapsto (v_1+v_2, v_2)$.

Can push forward 1-parameter subgroups by $\phi_*: \lambda \mapsto \phi\circ\lambda$.

There is an equivalence of categories \begin{align*} \begin{align*} \left\{{\text{Normal toric varieties}}\right\} &\rightleftharpoons\left\{{\text{lattice fans}}\right\}\\ (X, {\mathbb{T}}, x_0) &\mapsto (\Sigma(X), \Lambda({\mathbb{T}})) \\ (\phi, \tilde \phi) &\mapsto \tilde \phi_* \end{align*}\end{align*} i.e. send a toric variety to its fan convergency cones.