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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:


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 \(x_0\):


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.

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