Quick Criteria
Quick criteria:
- Normal \(\iff\) Saturated: For affines, \(X = \operatorname{Spec}{\mathbf{C}}[S]\) where \(S \subseteq M\) is a saturated semigroup. This is true for \(S = S_\sigma = \sigma {}^{ \vee }\cap M\) where \(\sigma\) is any SCRPC.
- Complete/proper \(\iff\) Full support: \(X_\Sigma\) is complete iff \(\mathop{\mathrm{supp}}\Sigma = N_{\mathbb{R}}\).
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Smooth \(\iff\) Lattice basis:
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For a cone \(\sigma = { \mathrm{Cone} }(S)\) is smooth iff \(\operatorname{det}S = \pm 1\), the volume of the standard lattice \({\mathbf{Z}}^n\).
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Consequences of smoothness:
- \(\operatorname{CDiv}(X) = \operatorname{Div}(X)\)
- \(\operatorname{Cl} (X) = \operatorname{Pic}(X)\)
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Consequences of smoothness:
- Smooth implies simplicial, so non-simplicial cones are singular.
- For \(p_\sigma\) the \(T{\hbox{-}}\)fixed point corresponding to \(\sigma\), \(T_p X \cong H\) where \(H\) is a Hilbert basis for \(S_\sigma\).
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For a cone \(\sigma = { \mathrm{Cone} }(S)\) is smooth iff \(\operatorname{det}S = \pm 1\), the volume of the standard lattice \({\mathbf{Z}}^n\).
- Simplicial \(\iff\) Euclidean basis: For \(\sigma = { \mathrm{Cone} }(S)\), \(\sigma\) is simplicial iff \(\operatorname{det}(S) \neq 0\).
- Orbifold singularities \(\iff\) Simplicial: \(X_\Sigma\) has at worst finite quotient singularities iff \(\Sigma\) is simplicial.
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Projectivity \(\iff\) Admits a strictly upper convex support function: For \(h\) a support function and \(D_h\) its associated divisor, the linear system \({\left\lvert {D_h} \right\rvert}\) defines an embedding \(X(\Delta) \hookrightarrow{\mathbf{P}}^N\) iff \(h\) is strictly upper convex.
- Alternatively, \(X_\Sigma\) is projective iff \(\Sigma\) arises as the normal fan of a polytope.
- Globally generated \(\iff\) Upper convex support function: \({\mathcal{O}}(D)\) is globally generated iff \(\psi_D\) is upper convex.
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Ample $
\iff
Strictly upper convex support function:
\(D\in \operatorname{CDiv}_T(X)\) is ample iff \(\psi_D\) is strictly upper convex. - \({\mathbf{Q}}{\hbox{-}}\)factorial \(\iff\) simplicial: iff every cone is simplicial.
- Global sections: for \(D\in \operatorname{Div}_T(X)\), \(P_D\) its associated polyhedron, \begin{align*} H^0(X; {\mathcal{O}}_X(D)) = \bigoplus _{m\in P_D \cap M} {\mathbf{C}}\, \chi^m .\end{align*}
Cones and Lattices
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Characters: for groups \(G\), a map \(\chi\in {\mathsf{Grp}}(G, {{\mathbf{C}}^{\times}})\). For \(G= T = ({{\mathbf{C}}^{\times}})^n\), there is an isomorphism \begin{align*} {\mathbf{Z}}^n & { \, \xrightarrow{\sim}\, }{\mathsf{Grp}}(T, {{\mathbf{C}}^{\times}}) \\ m = {\left[ {m_1,\cdots, m_n} \right]} &\mapsto \chi_m: {\left[ {t_1,\cdots, t_n} \right]} \mapsto \prod t_i^{m_i} .\end{align*} Generally set \(M \coloneqq{\mathsf{Grp}}(T, {{\mathbf{C}}^{\times}})\), the character lattice.
- \(M\) is a lattice, \(M_{\mathbb{R}}\coloneqq M\otimes_{\mathbf{Z}}{\mathbb{R}}\) is its associated Euclidean space.
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Cocharacters / one-parameter subgroups: for groups \(G\), a map \(\lambda \in {\mathsf{Grp}}({{\mathbf{C}}^{\times}}, G)\). For \(G = T = {{\mathbf{C}}^{\times}}\), there is again an isomorphism \begin{align*} {\mathbf{Z}}^n &\mapsto {\mathsf{Grp}}({{\mathbf{C}}^{\times}}, T) \\ u ={\left[ {u_1,\cdots, u_n} \right]} &\mapsto \lambda^u: t\mapsto {\left[ {t^{u_1}, \cdots, t^{u_n}} \right]} .\end{align*} Define \(N \coloneqq{\mathsf{Grp}}({{\mathbf{C}}^{\times}}, T)\) the cocharacter lattice.
- \(N\) is a lattice, \(N_{\mathbb{R}}\coloneqq N\otimes_{\mathbf{Z}}{\mathbb{R}}\) its associated euclidean space.
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There is a perfect pairing \begin{align*} {\left\langle {{-}},~{{-}} \right\rangle}: M\times N &\to {\mathbf{Z}}\\ ,\end{align*} defined using the fact that if \(m\in M, n\in N\) then \(\chi^m \circ \lambda^n \in {\mathsf{Grp}}({{\mathbf{C}}^{\times}}, {{\mathbf{C}}^{\times}})\) is of the form \(t\mapsto t^\ell\), so set \({\left\langle {m},~{n} \right\rangle} \coloneqq\ell\).
- Thus \(M = {\mathsf{Grp}}(M, {\mathbf{Z}})\) and \(N = {\mathsf{Grp}}(N, {\mathbf{Z}})\).
- How to recover the torus: \begin{align*} N \otimes_{\mathbf{Z}}{{\mathbf{C}}^{\times}}&\to T \\ u\otimes t &\mapsto \lambda^u(t) .\end{align*}
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\(\Delta\) is a fan, a collection of strongly convex rational polyhedral cones:
- Cone: \(0\in \sigma\) and \({\mathbb{R}}_{\geq 0} \sigma \subseteq \sigma\).
- Strongly convex: contains no nonzero subspace, i.e. no line through \(\mathbf{0} \in N_{\mathbb{R}}\). Equivalently, \(\dim \sigma {}^{ \vee }= n\).
- Rational: generated by \(\left\{{v_i}\right\} \subseteq N\), i.e. of the form \({ \mathrm{Cone} }(S)\) for \(S \subseteq N\).
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Dual cones: \begin{align*} \sigma {}^{ \vee }&\coloneqq\left\{{ u\in M {~\mathrel{\Big\vert}~}{\left\langle {u},~{v} \right\rangle} \geq 0 \,\,\forall v\in M_{\mathbb{R}}}\right\} .\end{align*}
- If \(\sigma {}^{ \vee }= \bigcap_{i=1}^s H_{m_i}^+\) for \(m_i \subseteq \sigma {}^{ \vee }\) then \(\sigma {}^{ \vee }= { \mathrm{Cone} }(m_1,\cdots, m_s)\).
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Hyperplanes and closed half-spaces: \begin{align*} H_m &\coloneqq\left\{{u\in N_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} = 0}\right\} \subseteq N_{\mathbb{R}}\\ H_m^+ &\coloneqq\left\{{u\in N_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} \geq 0}\right\} \subseteq N_{\mathbb{R}} .\end{align*}
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Face: \(\tau \leq \sigma\) is a face iff \(\tau\) is of the form \(\tau = H_m \cap\sigma\) for some \(m\in \sigma {}^{ \vee }\subseteq M_{\mathbb{R}}\).
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Facet: codimension one faces, \(\Sigma(n-1)\) where \(n\coloneqq\dim N\).
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Ray: dimension 1 faces, \(\Sigma(1)\).
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The semigroup of a cone: \begin{align*} S_\sigma &\coloneqq\sigma {}^{ \vee }\cap M = \left\{{ u\in M {~\mathrel{\Big\vert}~}{\left\langle {u},~{v} \right\rangle} \geq 0 \,\,\forall v\in \sigma }\right\} .\end{align*}
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The semigroup algebra of a semigroup: \begin{align*} {\mathbf{C}}[S] \coloneqq\left\{{\sum_{s\in S} c_s \chi^s {~\mathrel{\Big\vert}~}c_s \in {\mathbf{C}}, c_s = 0 { \text{a.e.} }}\right\}, \qquad \chi^{m_1}\cdot \chi^{m_2} \coloneqq\chi^{m_1 + m_2} .\end{align*}
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Simplicial: the generators can be extended to an \({\mathbb{R}}{\hbox{-}}\)basis of \(N_{\mathbb{R}}\). E.g. not simplicial:
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Smooth: the minimal generators can be extended to a \({\mathbf{Z}}{\hbox{-}}\)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 \({\mathfrak{m}}_p = \left\{{\chi^m {~\mathrel{\Big\vert}~}m\in S_ \sigma}\right\}\), and \({\mathfrak{m}}_p/{\mathfrak{m}}_p^2 = \left\{{\chi^m {~\mathrel{\Big\vert}~}m \text{ is indecomposable in } S_ \sigma}\right\}\). This exactly corresponds to a Hilbert basis.
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Facet: face of codimension 1.
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Edge: face of dimension 1. Note that facets = edges in \(\dim N = 2\).
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Saturated: \(S\) is saturated if for all \(k\in {\mathbb{N}}\setminus\left\{{0}\right\}\) and all \(m\in M\), \(km\in S \implies m\in S\). Any SCRPC is saturated.
- E.g. \(S = \left\{{(4,0), (3,1), (1,3), (0, 4)}\right\}\) is not saturated since \(2\cdot(2,2) = (4, 4) \in {\mathbb{N}}S\) but \((2,2)\not\in S\).
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Normalization: in the affine case, write \(X = \operatorname{Spec}{\mathbf{C}}[S]\) with torus character lattice \(M = {\mathbf{Z}}S\), take a finite generating set \(S'\), and set \(\sigma = { \mathrm{Cone} }(S') {}^{ \vee }\). Then \(\operatorname{Spec}{\mathbf{C}}[\sigma {}^{ \vee }\cap M]\to X\) is the normalization.
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Distinguished points: each strongly convex \(\sigma \leadsto \gamma_\sigma \in U_\sigma\) a unique point corresponding to the semigroup morphism \(m\mapsto \indic(m\in \sigma {}^{ \vee }\cap M)\), which is \(T{\hbox{-}}\)fixed iff \(\sigma\) is full-dimensional.
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Orbits: \({\mathrm{Orb}}( \sigma) = T. \gamma_\sigma\), and \(V(\sigma)\coloneqq{ \operatorname{cl}} {\mathrm{Orb}}( \sigma)\).
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Orbit-Cone correspondence: there is a correspondence \begin{align*} \left\{{\text{Cones } \sigma \in \Sigma}\right\} &\rightleftharpoons\left\{{T{\hbox{-}}\text{orbits in } X_\Sigma}\right\} \\ \sigma &\mapsto {\mathrm{Orb}}(\sigma) \coloneqq T.\gamma_{\sigma} = \left\{{\gamma: S_\sigma \to {\mathbf{C}}{~\mathrel{\Big\vert}~}\gamma(m) \neq 0 \iff m\in \sigma {}^{ \vee }\cap M}\right\} \cong {\mathsf{Grp}}(\sigma \cap M, {{\mathbf{C}}^{\times}}) ,\end{align*} where \(\dim {\mathrm{Orb}}( \sigma) = \operatorname{codim}_{N_{\mathbb{R}}} \sigma\), and \(\tau \leq \sigma \implies { \operatorname{cl}} {\mathrm{Orb}}(\tau) \supseteq{ \operatorname{cl}} {\mathrm{Orb}}( \sigma)\) and in fact \({ \operatorname{cl}} {\mathrm{Orb}}(\sigma) = {\textstyle\coprod}_{\tau\leq \sigma} { \operatorname{cl}} {\mathrm{Orb}}( \tau)\).
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Star: define \(N_\tau \coloneqq{\mathbf{Z}}\left\langle{\tau \cap N}\right\rangle\) and \(N(\tau){\mathbb{R}}\coloneqq N_{\mathbb{R}}/ (N_\tau)_{\mathbb{R}}\) and \(\mkern 1.5mu\overline{\mkern-1.5mu\sigma\mkern-1.5mu}\mkern 1.5mu\) for the image of \(\sigma\) under the quotient map, then \begin{align*} \mathrm{Star}(\tau) \coloneqq\left\{{\mkern 1.5mu\overline{\mkern-1.5mu\sigma \mkern-1.5mu}\mkern 1.5mu\subseteq N(\tau)_{\mathbb{R}}{~\mathrel{\Big\vert}~}\sigma\leq \tau }\right\} \subseteq N(\tau)_{\mathbb{R}} .\end{align*} This is always a fan, and \(V(\tau) = X_{\mathrm{Star}(\tau)}\).
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Star subdivision: for \(\sigma = { \mathrm{Cone} }(S)\) for \(S \coloneqq\left\{{u_1,\cdots, u_n}\right\}\), set \(u_0 \coloneqq\sum u_i\) and take \(\Sigma'(\sigma)\) defined as the cones generated by subsets of \(\left\{{u_0, u_1, \cdots, u_n}\right\}\) not containing \(S\). The star subdivision of \(\Sigma\) along \(\sigma\) is \(\Sigma^\star(\sigma) \coloneqq(\Sigma \setminus\left\{{ \sigma }\right\}) \cup\Sigma'( \sigma)\).
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Blowups: \(\phi: X_{\Sigma^\star(\sigma)}\to X_{\Sigma}\) is the blowup at \(\gamma_ \sigma\).
Divisors
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(Weil) divisor: \(\operatorname{Div}(X) = \left\{{\sum n_i V_i {~\mathrel{\Big\vert}~}V_i \subseteq X, \operatorname{codim}V_i = 1}\right\}\).
- \({\mathcal{O}}_X(D)\): the (coherent) sheaf associated to a Weil divisor \(D\).
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Cartier divisor: \(\operatorname{CDiv}(X) = H^0(X; {\mathcal{K}}_X^{\times}/{\mathcal{O}}_X^{\times})\), the quotient of rational functions by regular functions. For \(X\) normal, equivalently locally principal (Weil) divisors, so \(D \leadsto \left\{{(U_i, f_i)}\right\}\) where \({ \left.{{D}} \right|_{{U_i}} } = \operatorname{Div}(f_i)\).
- \({\mathbf{Q}}{\hbox{-}}\)Cartier divisor: A \({\mathbf{Q}}{\hbox{-}}\)divisor \(D =\sum n_i D_i\) with \(n_i\in {\mathbf{Q}}\) is \({\mathbf{Q}}{\hbox{-}}\)Cartier when \(mD\) is Cartier for some \(m\in {\mathbf{Z}}_{\geq 0}\).
- \({\mathbf{Q}}{\hbox{-}}\)factorial: every prime divisor is \({\mathbf{Q}}{\hbox{-}}\)Cartier.
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Ray divisors: every \(\rho\in \Sigma(1)\) defines a divisor \(D_\rho \coloneqq V(\rho) \coloneqq{ \operatorname{cl}} {\mathrm{Orb}}( \rho)\).
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Very Ample: \({\mathcal{L}}\) which defines a morphism into \({\mathbf{P}}H^0(X; {\mathcal{L}}) \cong {\mathbf{P}}^N\).
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Ample: \({\mathcal{L}}\) is basepoint free and some power \({\mathcal{L}}^n\) is very ample.
- \(D\) is (very) ample iff \({\mathcal{O}}_X(D)\) is (very) ample, i.e. \(D\) is ample iff \(nD\) is very ample for some \(n\).
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Upper convex: \(f(n_1 + n_2) \leq f(n_1) + f(n_2)\).
- Strictly upper convex: \(\sigma_1\neq \sigma_2 \implies f_{\sigma_1} \neq f_{\sigma_2}\).
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Linearly equivalent divisors: \(D_1\sim D_2 \iff D_1 - D_2 = \operatorname{Div}(f)\) for some \(f\).
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Complete linear systems: \({\left\lvert {D} \right\rvert} = \left\{{D'\in \operatorname{Div}(X) {~\mathrel{\Big\vert}~}D'\sim D}\right\}\).
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Support function: \(\phi: \mathop{\mathrm{supp}}\Sigma \to {\mathbb{R}}\) where \({ \left.{{\phi}} \right|_{{\sigma}} }\) is linear for each cone \(\sigma\).
- Integral with respect to \(N\) iff \(\phi(\mathop{\mathrm{supp}}\Sigma \cap N) \subseteq {\mathbf{Z}}\). Defines a set of integral support functions \(\operatorname{SF}(\Sigma, N)\).
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The class group complement exact sequence: for \(D_1,\cdots, D_n \in \operatorname{Div}(X)\) distinct, \begin{align*} {\mathbf{Z}}^n &\to \operatorname{Cl} (X) \twoheadrightarrow \operatorname{Cl} (X\setminus\cup D_i) \\ e_1 &\mapsto [D_i] .\end{align*}
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\({\mathcal{O}}_X(D)\) is the sheaf \begin{align*} U\mapsto \left\{{f\in {\mathcal{K}}(X)^{\times}(U) {~\mathrel{\Big\vert}~}\operatorname{Div}(f) + { \left.{{D}} \right|_{{U}} } \geq 0 \in \operatorname{Cl} (U) }\right\} .\end{align*} Then \(D\in \operatorname{CDiv}(X) \iff {\mathcal{O}}_X(D) \in \operatorname{Pic}(X)\).
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The toric class group exact sequence: \begin{align*} M &\to \operatorname{Div}_T(X) \twoheadrightarrow \operatorname{Cl} (X) \\ m &\mapsto \operatorname{Div}(\chi^m) = \sum_\rho {\left\langle {m},~{u_\rho} \right\rangle} [D_\rho] \end{align*} where \(u_\rho\) are minimal ray generators.
Polytopes
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Supporting hyperplanes: the positive side of an affine hyperplane \begin{align*} H_{u, b} &\coloneqq\left\{{m\in M_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} = b}\right\} \\ H_{u, b}^+ &\coloneqq\left\{{m\in M_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} \geq b}\right\} .\end{align*}
- If \(P\) is full dimensional and \(F\leq P\) is a facet, then \(F = P \cap H_{u_F, -a_F}\) for a unique pair \((u_F, a_F) \in N_{\mathbb{R}}\times {\mathbb{R}}\).
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Polytope: the convex hull of a finite set \(S \subseteq N_{\mathbb{R}}\) or an intersection of half-spaces: \begin{align*} P = \left\{{\sum_{v\in S} \lambda_v v {~\mathrel{\Big\vert}~}\sum \lambda_v = 1}\right\} = \bigcap_{i=1}^s H_{u_i, b_i}^+ .\end{align*}
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Simplex \(\dim P = d\) and there are exactly \(d+1\) vertices.
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Simple: \(\dim P = d\) and every vertex is the intersection of exactly \(d\) facets.
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Simplicial: all facets are simplices.
- E.g. simple but not simplicial: the cube in \({\mathbb{R}}^3\), since each vertex meets 3 edges but a square is not a simplex. -E.g. Simplicial but not simple: the octahedron in \({\mathbb{R}}^3\), since each vertex meets 4 edges but each face is a triangle.
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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.
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Polar dual: for \(P \subseteq M_{\mathbb{R}}\), \begin{align*} P^\circ = \left\{{u\in N_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} \geq - 1\,\, \forall m\in P}\right\} .\end{align*}
- Trick: for \(P \subseteq M_{\mathbb{R}}\) with \(0\in P\), \begin{align*} P = \left\{{m\in M_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u_F} \right\rangle} \geq -a_F,\, F \in \mathrm{Facets}(P) }\right\} \\ \implies P^\circ = \Conv(\left\{{ a_F^{-1}u_F }\right\}) \subseteq N_{\mathbb{R}} .\end{align*} E.g. write the square as \(\left\{{{\left\langle {m},~{\pm e_i} \right\rangle}\geq -1}\right\}\), then \(a_F = 1\) for all \(F\):
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Cone on a polytope: \(C(P) \coloneqq{ \mathrm{Cone} }(P\times \left\{{1}\right\}) \subseteq M_{\mathbb{R}}\times {\mathbb{R}}\), the set of cones through all proper faces of \(P\).
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Normal: \(\qty{kP \cap M} + \qty{\ell P \cap M} \subseteq (k+\ell)P \cap M\), or equivalently \(k\cdot (P \cap M) = (kP) \cap M\), or equivalently \((P \cap M)\times\left\{{1}\right\}\) generates \(C(P) \cap(M\times {\mathbf{Z}})\) as a semigroup.
- If \(P \subseteq M_{\mathbb{R}}\) 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.
- \(P\leadsto {\mathcal{L}}_P \in \operatorname{Pic}(X_P)\)
- \(P \cap M \leadsto H^0(X_P; {\mathcal{L}}_P)\).
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Reflexive: a polytope \(P\) with facet presentation \begin{align*} P = \left\{{m\in M_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{\mu_F} \right\rangle} \geq -1 \forall F\in \mathrm{Facets}(P)}\right\} .\end{align*} Implies that \(\int(P) \cap M = \left\{{\mathbf{0}}\right\}\), and \(P^\circ = \Conv(\left\{{u_F {~\mathrel{\Big\vert}~}F\in \mathrm{Facets}(P)}\right\})\).
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Polyhedron of a divisor \(P_D\): write \(D = \sum_{\rho} a_{\rho} D_{\rho}\), for any \(m\in M\), \(\operatorname{Div}(\chi^m) + D \geq 0 \implies {\left\langle {m},~{\rho} \right\rangle} \geq a_{\rho} \implies {\left\langle {m},~{\rho} \right\rangle} \geq - a_\rho\), so set \begin{align*} P_D \coloneqq\left\{{ m\in M_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{\rho } \right\rangle}\geq a_\rho \, \forall \rho \in \Sigma(1)}\right\} .\end{align*}
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Divisor of a polytope: \(D_P = \sum_F a_F D_F\) where \(P = \left\{{m {~\mathrel{\Big\vert}~}{\left\langle {m},~{u_F} \right\rangle} \geq -a_F}\right\}\).
- \(D_P\) is always the pullback of \({\mathcal{O}}_{{\mathbf{P}}^N}(1)\) along the embedding.
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Very ample polytopes: for every vertex \(v\), the semigroup \(\left\{{m' - v {~\mathrel{\Big\vert}~}m'\in P \cap M}\right\}\) is saturated in \(M\).
- Gives an embedding \(X \hookrightarrow{\mathbf{P}}^N\) where \(N = {\sharp}(P \cap M) - 1\).
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The toric variety of a polytope: if \(P \cap M = \left\{{m_1,\cdots, m_s}\right\}\) and \(P\) is full dimensional very ample, then writing \(T_N\) for the torus of \(N\), \begin{align*} X_{P} \coloneqq{ \operatorname{cl}} \operatorname{im}\phi,\qquad \phi: T_N &\to {\mathbf{P}}^{s-1} \\ t &\mapsto {\left[ {\chi^{m_1}(t) : \cdots : \chi^{m_s}(t)} \right]} .\end{align*}
- Vertices \(m_i\) correspond to \(U_{\sigma_i}\) for \(\sigma_i = { \mathrm{Cone} }(P \cap M - m_i) {}^{ \vee }\):
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Smooth: \(P\) is smooth iff for all vertices \(v\in P\), \(\left\{{w_E - v{~\mathrel{\Big\vert}~}E\text{ is an edge containing }v}\right\}\) can be extended to a \({\mathbf{Z}}{\hbox{-}}\)basis of \(M\), where \(w_E\) is the first lattice point on \(E\).
Singularities and Classification
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- Gorenstein: \(X\) normal where \(K_X \in \operatorname{CDiv}(X)\) 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.
Classification of smooth complete toric varieties:
- \(\dim \Sigma = 2, {\sharp}\Sigma(1) = 3\): 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 \(\mathop{\mathrm{supp}}\Sigma = {\mathbb{R}}^2\), and determinants for \({\left\lvert {a} \right\rvert} = {\left\lvert {b} \right\rvert} = 1\), so \((-1, 1)\).
- \(\dim \Sigma = 2, {\sharp}\Sigma(1) = 4\): 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 \(\operatorname{det}{ \begin{bmatrix} {-1} & {a} \\ {b} & {-1} \end{bmatrix} } = 1-ab = \pm 1 \implies ab=0,2\), so \((a,b) = (2,1), (1,2), (-2, -1), (-1,-2)\).
- \(\dim \Sigma = 2, {\sharp}\Sigma(1) = d\), smooth: \(\operatorname{Bl}_{p_1,\cdots, p_\ell} X\) for \(X = {\mathbf{P}}^2\) or \({ \mathbf{F} }_a\) for some \(a\) and \(p_i\) torus fixed points.
Examples
Things you can figure out for every example:
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Given \(\Delta\), for \(\sigma\in \Delta\),
- What is \(\sigma {}^{ \vee }\)?
- Generators for \(S_\sigma\)?
- Describe \(U_\sigma\) and \(X(\Delta)\).
- What are the transition functions for \(U_{\sigma_1} \to U_{\sigma_2}\) when \(\sigma_1 \cap\sigma_2 = \tau\) intersect in a common face?
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What are the \(T{\hbox{-}}\)invariant points?
- What are the \(T{\hbox{-}}\)invariant divisors \(D_{\rho_i}\)?
- What are all of the \(T{\hbox{-}}\)orbit closures of various dimensions?
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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 \(T{\hbox{-}}\)invariant point?
- What is the associated polytope \(P_\Delta\)? What is its polar dual \(P_\Delta^\circ\)?
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What are the intersection numbers \(D_{\rho_i} \cdot D_{\rho_j}\)?
- What are the self-intersection numbers \(D_{\rho_i}^2\)?
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What is \(\operatorname{Div}_T(X)\)? \(\operatorname{CDiv}_T(X)\)?
- Which divisors are ample? Very ample? Globally generated?
- What is \(\operatorname{Cl} (X)\)? \(\operatorname{Pic}(X)\)?
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What is \(K_X\)?
- Is \(K_X\) ample?
- Is \(X(\Delta)\) projective?
- What is \(H^0(X(\Delta); {\mathcal{O}}(D) )\) for \(D\in \operatorname{Div}_T(X)\)?
- What is the Poincaré polynomial of \(X(\Delta)\)? (I.e. what are the Betti numbers?)
Some useful explicit varieties:
- \(V(x^3-y^2)\) with torus \(T = \left\{{{\left[ {t^2, t^3} \right]} {~\mathrel{\Big\vert}~}t\in {{\mathbf{C}}^{\times}}}\right\}\).
- \(V(xy-zw)\) with torus \(T = \left\{{{\left[ {a,b,c,abc^{-1}} \right]} {~\mathrel{\Big\vert}~}a,b,c,d\in {{\mathbf{C}}^{\times}}}\right\}\).
- \(V(xz-y^2)\), note \(V(x, y)\in \operatorname{Div}(X) \setminus\operatorname{CDiv}(X)\).
- \(\Im([x:y] &\mapsto [x^3: x^2y : xy^2 : y^3])\) the twisted cubic. Corresponds to \(\sigma {}^{ \vee }= \left\{{(3,0), (2,1), (1,2), (0, 3)}\right\}\).
- 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 \({\left[ {s,t} \right]} &\mapsto {\left[ {1:s:s^2:t:st} \right]}\).
- The Segre variety: \(\operatorname{Spec}{\mathbf{C}}[x_1y_1, x_1 y_2, \cdots, x_1 y_n, x_2 y_1, \cdots, x_m y_1, \cdots x_m y_n]\).
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- \(({\mathbf{C}}^{\times})^n\): Take \(\Delta = \left\{{ \sigma_0 = {\mathbb{N}}\left\langle{0}\right\rangle}\right\} \subseteq N\) with \(\dim N = n\) yields \(S_{\sigma_0} = {\mathbb{N}}\left\langle{\pm e_1 {}^{ \vee },\cdots, \pm e_n {}^{ \vee }}\right\rangle = M\) for so \(X(\Delta) = \operatorname{Spec}{\mathbf{C}}[x_1^{\pm 1},\cdots, x_n^{\pm 1}] = ({\mathbb{G}}_m)^n\).
- \({\mathbf{C}}^n\): Take \(\Delta = { \mathrm{Cone} }(\sigma_0 = {\mathbb{N}}\left\langle{ e_1,\cdots, e_n}\right\rangle )\) yields the positive orthant \(S_{\sigma_0} = {\mathbb{N}}\left\langle{e_1 {}^{ \vee },\cdots, e_n {}^{ \vee }}\right\rangle \subseteq M\), so \(X(\Delta) = \operatorname{Spec}{\mathbf{C}}[x_1,\cdots, x_n] = {\mathbf{A}}^n\).
- The quadric cone: \(\Delta = { \mathrm{Cone} }(\sigma_1 = {\mathbb{N}}\left\langle{e_2, 2e_1 - e_2}\right\rangle)\) yields \(S_{\sigma_1} = {\mathbb{N}}\left\langle{e_1 {}^{ \vee }, e_1 {}^{ \vee }+ e_2 {}^{ \vee }, e_1 {}^{ \vee }+ 2e_2 {}^{ \vee }}\right\rangle\) so \(X(\Delta) = \operatorname{Spec}{\mathbf{C}}[x, xy, xy^2] = \operatorname{Spec}{\mathbf{C}}[u,v,w]/(v^2-uw)\):
- \({\mathbf{P}}^1\): Take \(\Delta = \left\{{{\mathbb{R}}_{\geq 0}, 0, {\mathbb{R}}_{\leq 0}}\right\}\) and glue along overlaps to get \(X(\Delta) = {\mathbf{P}}^1\) with gluing maps \(x\mapsto x^{-1}\):
- \(\operatorname{Bl}_1 {\mathbf{C}}^2\): Take \(\sigma_0 = {\mathbb{N}}\left\langle{ e_2, e_1+e_2}\right\rangle\) and \(\sigma_1 = {\mathbb{N}}\left\langle{e_1+e_2, e_1}\right\rangle\) to get \(U_{ \sigma_0} = \operatorname{Spec}{\mathbf{C}}[x, x^{-1}y]\) and \(U_{ \sigma_1} = \operatorname{Spec}{\mathbf{C}}[y, xy^{-1}]\), both copies of \({\mathbf{C}}^2\):
Why this is a blowup of \({\mathbf{C}}^2\): write \(\operatorname{Bl}_1 {\mathbf{C}}^2 = V(xt_1 - yt_0) \subseteq {\mathbf{C}}^2\times {\mathbf{P}}^1\) for \({\mathbf{P}}^1 = \left\{{{\left[ {t_0: t_1} \right]}}\right\}\). Take the open cover \(U_i = D(t_i) \cong {\mathbf{C}}^2\), where coordinates on \(U_0\) are \(x, t_1/t_0 = x^{-1}y\) and on \(U_1\) are \(y, t_0/t_1 = xy^{-1}\) and glue.
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\({\mathbf{P}}^2\): take \(\Delta = { \mathrm{Cone} }(e_1, e_2, -e_1-e_2)\):
This has dual cone:
Each \(U_{\sigma_i} \cong {\mathbf{C}}^2\) with coordinates \((x,y), (x^{-1}, x^{-1}y), (y^{-1}, xy^{-1})\) respectively for \(U_i\). Glue to obtain \(x=t_1/t_0, y=t_2/t_0\).
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\(F_a\) the Hirzebruch surface: take \({ \mathrm{Cone} }(e_1, -e_2, -e_1, -e_1 + ae_2)\) to get
- \(U_{\sigma_1} = \operatorname{Spec}{\mathbf{C}}[x,y]\),
- \(U_{\sigma_2} = \operatorname{Spec}{\mathbf{C}}[x,y^{-1}]\),
- \(U_{\sigma_3} = \operatorname{Spec}{\mathbf{C}}[x^{-1},x^{-a} y^{-1}]\),
- \(U_{\sigma_4} = \operatorname{Spec}{\mathbf{C}}[x^{-1},x^a y]\),
which patch in the following way:
Project to \(y=0\) to get the patching \(x\mapsto x^{-1}\), so a copy of \({\mathbf{P}}^1\). Patching in the fiber direction, e.g. \(U_{\sigma_1}\) and \(U_{\sigma_2}\), gives a copy of \({\mathbf{C}}\times {\mathbf{P}}^1\). Thus this is a bundle \({\mathbf{P}}^1\to {\mathcal{E}}\to {\mathbf{P}}^1\).
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\({\mathbf{C}}\times {\mathbf{P}}^1\): todo.
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\({\mathbf{P}}^1 \times {\mathbf{P}}^1\): todo.
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\({\mathbf{C}}^a \times {\mathbf{P}}^b\): todo.
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\({\mathbf{P}}^a \times {\mathbf{P}}^b\): todo.
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This thing:
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\(({\mathbf{P}}^2, {\mathcal{O}}(1))\): take \(P = \Conv(0, e_1, e_2)\), so \(X_P = { \operatorname{cl}} \Phi_P\) where \begin{align*} \Phi_P: ({{\mathbf{C}}^{\times}})^2 &\to {\mathbf{P}}^2 \\ (s,t) &\mapsto [1: s: t] ,\end{align*} which is the identity embedding corresponding to \({\mathcal{O}}(1)\) on \({\mathbf{P}}^2\).
- \(2P\) yields \begin{align*} \Phi_{2P}: ({{\mathbf{C}}^{\times}})^2 &\to {\mathbf{P}}^5 \\ (s,t) &\mapsto [1: s: t : s^2: st: t^2] ,\end{align*} the Veronese embedding corresponding to \({\mathcal{O}}(2)\) on \({\mathbf{P}}^2\).
Some useful facts about \({\mathbf{P}}^n\):
- The torus embedding is \begin{align*} ({{\mathbf{C}}^{\times}})^n &\hookrightarrow{\mathbf{P}}^n \\ {\left[ {a_1,\cdots, a_n} \right]} &\mapsto {\left[ {1: a_1 : \cdots : a_n} \right]} .\end{align*}
- The torus action is \begin{align*} ({{\mathbf{C}}^{\times}})^n &\curvearrowright{\mathbf{P}}^n \\ {\left[ {t_1,\cdots, t_n} \right]} . {\left[ {x_0: x_1:\cdots:x_n} \right]} &= {\left[ {x_0: t_1 x_1:\cdots:t_n x_n} \right]} .\end{align*}