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# 2022-06-20

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 $$v_2 = {\left[ {1,2} \right]}$$: 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 $$\sigma {}^{ \vee }= \left\{{u\in U {~\mathrel{\Big\vert}~}{ \left.{{u}} \right|_{{\sigma}} } \geq 0}\right\}$$: 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: Fans in $${}_{{\mathbf{Q}}}{\mathsf{Mod}} ^{\mathrm{fg}}$$: a finite set of pointed cones closed under taking faces and intersections.

Example: 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 $$x_0$$: 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.