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Toric Varieties 1
Basics
Cones
Fans
Tori
Divisors
Link to Diagram - \begin{align*} \end{align*} Show that if \(X\) is an affine toric variety then \({\operatorname{Pic}}(X) = 0\). - \begin{align*} \end{align*} Give an characterization of when an invariant divisor is Cartier. - \begin{align*} \end{align*} Spoiler: \(D\) is Cartier iff there exists an \(m_\sigma\in M\) such that \({ \left.{{D}} \right|_{{X_\sigma}} } = { \left.{{ \operatorname{Div}(\chi^{m_\sigma})}} \right|_{{X_\Sigma}} }\). - \begin{align*} \end{align*} What is the poset structure on \(\Sigma\)? - \begin{align*} \end{align*} What is the inverse limit characterization of \(\mathop{\mathrm{Cart}}\operatorname{Div}_{T_N}(X_\Sigma)\)? - \begin{align*} \end{align*} Spoiler: \(\mathop{\mathrm{Cart}}\operatorname{Div}_{T_N}(X_\Sigma) = \cocolim M/M(\sigma)\) over the face poset. - \begin{align*} \end{align*} What is the lattice point associated to an invariant Cartier divisor \(D\in\mathop{\mathrm{Cart}}\operatorname{Div}_{T_N}(X_\Sigma)\)? - \begin{align*} \end{align*} What is the associated linear functional and associated Weyl divisor? - \begin{align*} \end{align*} Spoiler: \({\left\langle {m_\sigma},~{{-}} \right\rangle}\), which takes the value \(a_\rho\) for a ray. Associate \(D \coloneqq\sum_{\rho \in \Sigma(1)} a_\rho D_\rho\). - \begin{align*} \end{align*} What is the support function associated to \(D\in \mathop{\mathrm{Cart}}\operatorname{Div}_{T_N}(X_\Sigma)\)? - \begin{align*} \end{align*} What is the polyhedron associated to a Weil divisor \(D\)? - \begin{align*} \end{align*} Spoiler: \(P_{D}=\left\{x \in M_{\mathrm{R}}:\left\langle m, u_{\rho}\right\rangle \geq-a_{\rho}, \quad \forall \rho \in \Sigma(1)\right\}\). - \begin{align*} \end{align*} Describe the bijection between \({{\Gamma}\qty{X_\Sigma; {\mathcal{O}}(D)} }\) and lattice points in \(P_D\). - \begin{align*} \end{align*} Spoiler: \({{\Gamma}\qty{X_\Sigma; {\mathcal{O}}(D)} } = \bigoplus_{m\in P_d} {\mathbb{C}}\left\langle{\chi^{m}}\right\rangle\), and the number of lattice points is the dimension \(h^0(X; {\mathcal{O}}(D))\). - \begin{align*} \end{align*} Describe how to present a polytope in terms of intersecting half-spaces. - \begin{align*} \end{align*} Spoiler:
- \begin{align*} \end{align*} Describe the Weil divisor associated to the facets. Why is it also Cartier? - \begin{align*} \end{align*} Describe the bijection between vertices of \(P\) and maximal cones in \(\Sigma_P\). - \begin{align*} \end{align*} For which \(k\) is \(kD_P\) very ample? - \begin{align*} \end{align*} Give a characterization for when an invariant Cartier divisor \(D\) on \(X_{\Sigma(P)}\) is ample. - \begin{align*} \end{align*} Spoiler: iff \(P_D\) is a lattice polytope with the same normal fan as \(P\). - \begin{align*} \end{align*} What is a convex function - \begin{align*} \end{align*} Spoiler: \(\psi(t u+(1-t) v) \geq t \psi(u)+(1-t) \psi(v)\). - \begin{align*} \end{align*} What does it mean for a divisor \(D\) to be basepoint-free? - \begin{align*} \end{align*} Spoiler: generated by global sections. - \begin{align*} \end{align*} Give several characterizations for when an invariant Cartier divisor \(D\) is basepoint-free. - \begin{align*} \end{align*} Spoiler: iff the support function \(\psi_D\) is a convex function, iff \(m_\sigma\in P_D\) for all \(\sigma \in \Sigma(n)\), iff \(P_D = \Conv\left\{{m_\sigma {~\mathrel{\Big\vert}~}\sigma\in \Sigma(n)}\right\}\) (the convex hull). - \begin{align*} \end{align*} Give a characterization for when such a \(D\) is ample. - \begin{align*} \end{align*} Spoiler: iff \(\psi_D\) is strictly convex.
Polytopes
Examples
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Projective space:
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- Spoiler: \(S_{\sigma} = \left\langle{e_{1}, e_{1}+e_{2}, e_{2}+e_{3}, e_{3}}\right\rangle_{{\mathbb{N}}}\) \({\mathbb{C}}[\sigma] = {\mathbb{C}}[x,xy,yz,z] = {\mathbb{C}}[x,y,z,w]/\left\langle{xz-yw}\right\rangle\), and \(X_\sigma = V(xz-yw)\). This is singular at the origin but not an orbifold since it is not simplicial (the singularities are worse).
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Spoiler: \(\operatorname{Div}\left(\chi^{u_{1}}\right)=\left\langle u_{1}, u_{0}\right) D_{0}+\left\langle u_{1}, u_{1}\right\rangle D_{1}+\left\langle u_{1}, u_{2}\right\rangle D_{2}\), \(\operatorname{Div}(\chi^{u_2}) =D_2 + D_0\), and
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Spoiler: \(\operatorname{Div}\left(\chi^{u_{1}}\right)=\left\langle u_{1}, u_{0}\right) D_{0}+\left\langle u_{1}, u_{1}\right\rangle D_{1}+\left\langle u_{1}, u_{2}\right\rangle D_{2}\), \(\operatorname{Div}(\chi^{u_2}) =D_2 + D_0\), and
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Monoids/Semigroups
- What is a semigroup?
- What is a monoid?
- What is a saturated monoid?
- Show that a cuspidal curve corresponds to \(k[x^2, x^3] = 1 \oplus x^2 \oplus x^3 \oplus x^4 \oplus \cdots\)
- Show that \({\mathbb{A}}^1\) corresponds to \(k[{\mathbb{N}}]\).