--- aliases: - Toric Varieties 1 date: "2022-06-03 22:28" modification date: "Friday 3rd June 2022 22:28:36" title: Toric Varieties 1 --- Last modified date: \<%+ tp.file.last_modified_date() %\> ------------------------------------------------------------------------ - Tags: - #todo/untagged - Refs: - - Links: - [toric variety](Unsorted/toric.md) - [examples of varieties](Unsorted/examples%20of%20varieties.md) ------------------------------------------------------------------------ # Toric Varieties 1 ## Basics - [ ] What is a toric variety? - [ ] What is a torus? - [ ] What is a normal variety? - [ ] What is the relevant commuting diagram for defining a toric variety in terms of a torus action? - [ ] What is a [separated](Unsorted/separated.md) variety? - [ ] What is an abstract variety? - [ ] What is weighted projective space? - [ ] When does a sequence of vectors in ${\mathbb{Z}}^n$ generate ${\mathbb{R}}^n$ as a lattice? - [ ] Show that $\operatorname{coker}(V) {}^{ \vee }\cong \ker(V^t)$. - [ ] Show that $A {}^{ \vee }\otimes B { \, \xrightarrow{\sim}\, }\mathop{\mathrm{Hom}}(A, B)$. - [ ] What is the Segre embedding? - [ ] What is the Veronese embedding? - [ ] Spoiler: Veronese is the image of the map given by all monomials of a given degree. Segre: a product of $d$ projective spaces, take monomials of degree $d$ with one variable from each factor. ## Cones - [ ] What is a cone? - [ ] For a set of vectors $\left\{{v_i}\right\}_{i\leq m}$, what is ${ \mathrm{Cone} }(v_1,\cdots, v_m)$? - [ ] Show that if a point is in a cone, its entire ray is in the cone. - [ ] If $\sigma$ is a rational cone in $N_{\mathbb{R}}$, what is the dual rational cone $\sigma {}^{ \vee }$? - [ ] Spoiler: $\sigma {}^{ \vee }= \left\{{m \in N {}^{ \vee }_{\mathbb{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{s} \right\rangle} \geq 0 \,\,\forall s\in \sigma}\right\}$. - [ ] Draw $\sigma = { \mathrm{Cone} }(e_2, 2e_1-e_2)$ and $\sigma' = { \mathrm{Cone} }(e_1, e_1 + 2e_2)$. Show that $\sigma ' = \sigma {}^{ \vee }$. - [ ] Spoiler: ![](attachments/Pasted%20image%2020220605132839.png) - [ ] Draw $\sigma = { \mathrm{Cone} }(e_1)$ and $\sigma {}^{ \vee }$. - [ ] Spoiler: $\sigma {}^{ \vee }= { \mathrm{Cone} }(e_1, \pm e_2)$. - [ ] What is a strictly convex cone? - [ ] Spoiler: doesn't contain a linear subspace $L$ with $\dim L \geq 1$. - [ ] What is a full-dimensional cone? - [ ] Show that $\sigma$ is strictly convex iff $\sigma {}^{ \vee }$ is full-dimensional. - [ ] What is the convex polyhedral cone $\sigma$? - [ ] What does it mean for $\sigma$ to be rational? - [ ] What is the dimension of a cone? - [ ] What is the dual cone $\sigma {}^{ \vee }$? - [ ] What does it mean for $\sigma$ to be strongly convex? - [ ] What is a smooth cone? - [ ] Spoiler: $\sigma$ is smooth iff one can choosen generators of $\sigma$ which extend to a basis of $N$. - [ ] Show that $\sigma \coloneqq{ \mathrm{Cone} }(v_1,v_2)\subseteq {\mathbb{R}}^2$ is smooth iff $v_1,v_2\in {\mathbb{Z}}^2$ iff ${\left\lvert { \operatorname{det}{\left[ {v_1, v_2} \right]}^t} \right\rvert} = 1$. - [ ] What is a simplicial cone? - [ ] Spoiler: $\sigma$ is simplicial iff it can be generated by linearly independent vectors. - [ ] Show that smooth cones are simplicial. - [ ] Show that $\sigma$ is simplicial iff $X_\sigma$ is an orbifold. - [ ] What is a lattice? - [ ] What is its dual lattice? - [ ] What is a ray? - [ ] What is a ray generator? - [ ] What is a face $\tau$ of $\sigma$? - [ ] Spoiler: the intersection of a linear space with $\sigma$. - [ ] What is a facet? - [ ] What is a ray? - [ ] What is the orbit-cone correspondence? - [ ] Spoiler: $T{\hbox{-}}$orbits in $X$ correspond to faces in $\sigma$, matching $F \subset \sigma$ with $\left\{{x\in X {~\mathrel{\Big\vert}~}x^a = 0 \iff a\not\in F}\right\}$. - [ ] What is a strongly convex rational polyhedral cone (*scrapc*)? - [ ] What is a smooth scrapc? - [ ] What is a simplicial scrapc? - [ ] Show that if $\sigma = \left\langle{v_i}\right\rangle$ then ${\mathbb{C}}[S_\sigma] = {\mathbb{C}} { \left[ \scriptstyle {\chi^{v_i}} \right] } $ is generated by the corresponding characters. - [ ] For $\sigma$ a cone, what is the semigroup $S_\sigma$? - [ ] What is a semigroup? - [ ] Show that $X_\sigma \coloneqq\operatorname{Spec}{\mathbb{C}}[S_\sigma]$ is an affine toric variety. - [ ] Show that $X_\sigma$ is normal. - [ ] Show that if $X_\sigma$ is associated to the lattice $N$, then the torus of $X_\sigma$ is $T_N \coloneqq N\otimes_{\mathbb{Z}}({\mathbb{C}}^{\times})^n$. - [ ] Show that the ${\mathbb{C}}{\hbox{-}}$points of $X_\sigma$ are the semigroup morphisms $S_\sigma \to {\mathbb{C}}$, and similarly for $T_N$, $S_\sigma\to {\mathbb{C}}^{\times}$. - [ ] Describe a distinuished $T{\hbox{-}}$invariant point $x_\sigma\in X_\sigma$. ## Fans - [ ] What is a [fan](Unsorted/toric.md)? - [ ] What is a complete fan? - [ ] What is a rational polyhedral fan? - [ ] Spoiler: A rational polyhedral (r. p.) fan $\Sigma$ is a finite collection of r.p. (strictly convex) cones in $N_{\mathrm{R}}$ which intersect along a common face and which contains all faces of the cones in the collection. - [ ] What is a fan for ${ \operatorname{Tot} }({\mathcal{O}}_{{\mathbb{P}}^1}(n))$? - [ ] See - [ ] Show that if the fan $\Sigma$ of $X_{\Sigma}$ has all its primitive generators lying on a hyperplane of $N_{\mathbb{R}}$, then $X_{\Sigma}$ is Calabi-Yau. - [ ] See - [ ] Describe how to associate to a cone an affine variety. - [ ] Descrbe how to associate an (abstract) variety to a fan by gluing varieties for cones. - [ ] What is $X_\Sigma$? - [ ] When is $X_\Sigma$ a smooth variety? - [ ] Spoiler: iff each maximal cone is smooth. - [ ] When is $X_\Sigma$ an orbifold? - [ ] Spoiler: iff each maximal cone is simplicial. - [ ] When is $X_\Sigma$ proper/compact? - [ ] Spoiler: iff $\displaystyle\bigcup\sigma_i$ is equal to $N_{\mathbb{R}}$ - [ ] When is $X_\Sigma$ projective? - [ ] How is a blowup reflected in a fan? - [ ] Spoiler: sub-divide a cone. - [ ] How do you desingularize a toric surface? - [ ] Spoiler: subdivide cones until the determinants of any two consecutive faces is 1. - [ ] How is $\Sigma$ recovered from $X_\Sigma$? - [ ] Spoiker: if $\mathbf{u}\in N$ then $\lim_{t\to 0}{\left[ {t^{u_1}, \cdots, t^{u_n}} \right]}$ exists in $X_\Sigma$ iff $u\in \sigma$.\ - [ ] What is the correspondence between rays $\rho_i$ and torus-invariant divisors $D_i$? - [ ] How do toric varieties provide partial compactifications of their tori? - [ ] Describe the correspondence between toric morphisms $X_{\Sigma} \to X_{\Sigma'}$ and $(\Sigma, \Sigma'){\hbox{-}}$invariant morphisms of lattices $N\to N'$. ## Tori - [ ] What is an algebraic torus? - [ ] What are some examples of algebraic tori? - [ ] What is the coordinate ring of a torus? - [ ] What is the lattice of one-parameter subgroups of a torus? - [ ] What is a character of a torus? - [ ] Spoiler: Laurent monomials. Thus characters are points in $M \cong ({\mathbb{Z}}^n, +)$ where the group structure on $M$ is pointwise multiplication. - [ ] What are all of the characters of the standard torus? - [ ] What is the character lattice of a torus? - [ ] What are some examples of fans? - [ ] Spoiler: see [[examples of fans]] - [ ] Show that $T{\hbox{-}}$invariant subvarieties of $X_\Sigma$ biject with cones in $\Sigma$. ## Divisors - [ ] What is a prime [[divisor]]? - [ ] What is ${\mathcal{O}}_{X, D}$? - [ ] ![](attachments/Pasted%20image%2020220603222901.png) - [ ] Show that there is a bijection between prime divisors of $X = \operatorname{Spec}R$ and codimension 1 prime ideals in $R$. - [ ] Show that if a prime divisor $D$ corresponds to $p\in \operatorname{Spec}R$ then ${\mathcal{O}}_{X, D} = R \left[ { \scriptstyle { {p}^{-1}} } \right] \subseteq \operatorname{ff}(R)$. - [ ] Describe how to associate a rational function to a Weil divisor for a normal variety. - [ ] What is the twisted sheaf ${\mathcal{O}}_X(D)$? - [ ] Spoiler: ![](attachments/Pasted%20image%2020220603225645.png) - [ ] What is a ${\mathbb{Q}}{\hbox{-}}$Cartier divisor? - [ ] ![](attachments/Pasted%20image%2020220604164718.png) - [ ] What is the invariant divisor associated to a cone? - [ ] Spoiler: the closure of the corresponding torus orbit - [ ] Given a face $\tau \in \sigma_1 \cap\sigma_2$, describe how to glue $X_{\sigma_1}$ to $X_{\sigma_2}$. - [ ] Spoiler: in the semigroup, invert $m$, the lattice point where they intersect. ![](attachments/Pasted%20image%2020220605143149.png) - [ ] Describe the divisor map ${\mathbb{C}}(X_\Sigma) \to \operatorname{Div}(X_\Sigma)$. Why does it make sense to $\operatorname{Div}(\chi^m)$ for $\chi^m$ a character of $T_N$? - [ ] Give a formula for the toric boundary of $X_\Sigma$ in terms of divisors. - [ ] Spoiler: ![](attachments/Pasted%20image%2020220605164138.png) - [ ] Give a formula for $\operatorname{Div}(\chi^m)$. - [ ] Spoiler: ![](attachments/Pasted%20image%2020220605163001.png) - [ ] Describe to torus action on $\operatorname{Div}(X_\Sigma)$. - [ ] Spoiler: $t \cdot \sum a_{D} D \mapsto \sum a_{D}tD$ - [ ] Describe the torus-invariant divisors. - [ ] Spoiler: $\operatorname{Div}_{T_{N}}\left(X_{\Sigma}\right)=\left\{\sum_{\rho \in \Sigma(1)} a_{\rho} D_{\rho}\right\}$. - [ ] Describe the exact sequence involving $\operatorname{Div}_{T_N}(X_\Sigma)$. When is this short exact? - [ ] Spoiler: let $\left\{{\rho_i}\right\}$ be generating rays, then $\ker f = 0$ when the rays generate ${\mathbb{R}}^n$: `

`{=html} > [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMiwyLCJNIl0sWzQsMiwiXFxEaXZfe1RfTn0oWF9cXFNpZ21hKSJdLFs2LDIsIlxcQ2woWF9cXFNpZ21hKSJdLFs4LDIsIjAiXSxbMiwzLCJtIl0sWzQsMywiXFx0dntcXGlucHttfXtcXHJob18xfSwgXFxjZG90cywgXFxpbnB7bX17XFxyaG9fcn19Il0sWzIsMCwiXFxaWl5uIl0sWzQsMCwiXFxaWl57XFxzaXplIFxcU2lnbWEoMSl9Il0sWzAsMiwiXFxrZXIgZiJdLFsxLDIsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFswLDEsImYiXSxbNCw1LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzgsMF0sWzYsMCwiXFxjb25nIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dLFs3LDEsIlxcY29uZyIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XV0=) - [ ] What is the exact sequence for Cartier divisors? When is it short exact? - [ ] Spoiler: The following, which is exact if $X_\Sigma$ contains no toric factors: `

`{=html} > [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMiwwLCJNIl0sWzQsMCwiXFxDYXJ0XFxEaXYoWF9cXFNpZ21hKSJdLFs2LDAsIlxcUGljKFhfXFxTaWdtYSkiXSxbOCwwLCIwIl0sWzIsMSwibSJdLFs0LDEsIlxcRGl2KFxcY2hpXm0pIl0sWzAsMCwiXFxrZXIgZiJdLFs2LDBdLFswLDFdLFsxLDJdLFsyLDNdLFs0LDUsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=) - \[ \] Show that if $X$ is an affine toric variety then ${\operatorname{Pic}}(X) = 0$. - \[ \] Give an characterization of when an invariant divisor is Cartier. - \[ \] Spoiler: $D$ is Cartier iff there exists an $m_\sigma\in M$ such that ${ \left.{{D}} \right|_{{X_\sigma}} } = { \left.{{ \operatorname{Div}(\chi^{m_\sigma})}} \right|_{{X_\Sigma}} }$. - \[ \] What is the poset structure on $\Sigma$? - \[ \] What is the inverse limit characterization of $\mathop{\mathrm{Cart}}\operatorname{Div}_{T_N}(X_\Sigma)$? - \[ \] Spoiler: $\mathop{\mathrm{Cart}}\operatorname{Div}_{T_N}(X_\Sigma) = \cocolim M/M(\sigma)$ over the face poset. - \[ \] What is the lattice point associated to an invariant Cartier divisor $D\in\mathop{\mathrm{Cart}}\operatorname{Div}_{T_N}(X_\Sigma)$? - \[ \] What is the associated linear functional and associated Weyl divisor? - \[ \] Spoiler: ${\left\langle {m_\sigma},~{{-}} \right\rangle}$, which takes the value $a_\rho$ for a ray. Associate $D \coloneqq\sum_{\rho \in \Sigma(1)} a_\rho D_\rho$. - \[ \] What is the support function associated to $D\in \mathop{\mathrm{Cart}}\operatorname{Div}_{T_N}(X_\Sigma)$? - \[ \] What is the polyhedron associated to a Weil divisor $D$? - \[ \] Spoiler: $P_{D}=\left\{x \in M_{\mathrm{R}}:\left\langle m, u_{\rho}\right\rangle \geq-a_{\rho}, \quad \forall \rho \in \Sigma(1)\right\}$. - \[ \] Describe the bijection between ${{\Gamma}\qty{X_\Sigma; {\mathcal{O}}(D)} }$ and lattice points in $P_D$. - \[ \] 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 $h^0(X; {\mathcal{O}}(D))$. - \[ \] Describe how to present a polytope in terms of intersecting half-spaces. - \[ \] Spoiler: ![](attachments/Pasted%20image%2020220605175841.png) - \[ \] Describe the Weil divisor associated to the facets. Why is it also Cartier? - \[ \] Describe the bijection between vertices of $P$ and maximal cones in $\Sigma_P$. - \[ \] For which $k$ is $kD_P$ very ample? - \[ \] Give a characterization for when an invariant Cartier divisor $D$ on $X_{\Sigma(P)}$ is ample. - \[ \] Spoiler: iff $P_D$ is a lattice polytope with the same normal fan as $P$. - \[ \] What is a convex function - \[ \] Spoiler: $\psi(t u+(1-t) v) \geq t \psi(u)+(1-t) \psi(v)$. - \[ \] What does it mean for a divisor $D$ to be basepoint-free? - \[ \] Spoiler: generated by global sections. - \[ \] Give several characterizations for when an invariant Cartier divisor $D$ is basepoint-free. - \[ \] 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). - \[ \] Give a characterization for when such a $D$ is ample. - \[ \] Spoiler: iff $\psi_D$ is strictly convex. ## Polytopes - [ ] What is a lattice polytope? - [ ] Spoiler: ![](attachments/Pasted%20image%2020220605161708.png) - [ ] What is a face of a lattice polytope? - [ ] Spoiler: ![](attachments/Pasted%20image%2020220605162155.png) - [ ] What is the difference between a polyhedron and a polytope? - [ ] What is the normal fan of a lattice polytope? - [ ] Describe the cone-face correspondence. - [ ] Find the normal fan of a triangle. - [ ] Spoiler: ![](attachments/Pasted%20image%2020220605162411.png) ## Examples - Projective space: - [ ] What is ${\mathbb{P}}^{n-1}$ as a quotient? What is $Z$, the primitive collection, $G$, and its action? - [ ] Spoiler: primitive collection $\left\{{e_0, \cdots, e_n}\right\}, Z= \left\{{\mathbf{0}}\right\}, G = {\mathbb{Z}}\otimes_{\mathbb{Z}}{\mathbb{C}}^{\times}= {\mathbb{Z}}$ with action $\lambda.{\left[ {x_0: \cdots: x_n} \right]} = {\left[ {\lambda x_0: \cdots : \lambda x_n} \right]}$. - [ ] What is the torus $T({\mathbb{P}}^n)$ in ${\mathbb{P}}^n$? - [ ] Characterize when two points of ${\mathbb{P}}^n$ are in the same torus orbit. - [ ] Show that $T({\mathbb{P}}^n)$ is the orbit of ${\left[ {1:1:\cdots:1} \right]}$. - [ ] What are the fixed points of the $T({\mathbb{P}}^n)$ action? - [ ] Construct ${\mathbb{P}}^2 = X_\Sigma$ for some fan $\Sigma$. Compute $X_{\sigma_i}$ for each cone $\sigma_i$, and show each corresponds to the open charts $U_i \coloneqq\left\{{x_i\neq 0}\right\}$. - [ ] Spoiler: take $\Sigma_1 = \left\{{e_1, e_2, v_0 \coloneqq-e_1-e_2 }\right\}$, or more generally any three vectors summing to zero which pairwise have determinant 1. ![](attachments/Pasted%20image%2020220605143625.png) - [ ] Orthants: show that $\sigma \coloneqq\sum {\mathbb{R}}_{\geq 0} e_i$ is self-dual and $X_\sigma = {\mathbb{C}}^n$. - [ ] Zero: show that $\sigma \coloneqq\left\{{0}\right\}$ satisfies $\sigma {}^{ \vee }= ({\mathbb{R}}^n) {}^{ \vee }$ and $X_\sigma = ({\mathbb{C}}^{\times})^n$. - [ ] Show that if $\sigma = {\mathbb{R}}_{\geq 0} e_1 \subseteq {\mathbb{R}}^2$ is the right half-ray then $X_\sigma = {\mathbb{C}}\times{\mathbb{C}}^{\times}$. - [ ] Show that $\sigma \coloneqq{ \mathrm{Cone} }(e_1, 2e_1-e_2)$ is simplicial but not smooth, - [ ] Find $\sigma {}^{ \vee }$ - [ ] Spoiler: $\sigma {}^{ \vee }= {\mathbb{R}}_{\geq 0}$, - [ ] Find $S_\sigma$, and show it can't be generated by only two vectors. - [ ] Spoiler: $S_\sigma = {\mathbb{Z}}_{\geq 0}\left\langle{e_1, e_1+e_2,e_1 + 2e_2}\right\rangle$ - [ ] Find the semigroup ring ${\mathbb{C}}[S_\sigma]$ and describe $X_\sigma$. - [ ] Spoiler: ${\mathbb{C}}[S_\sigma] = {\mathbb{C}} { \left[ \scriptstyle {x,xy,xy^2} \right] } = {\mathbb{C}} { \left[ \scriptstyle {x,y,z} \right] } /\left\langle{y^2-xz}\right\rangle$. - [ ] Where is $X_\sigma$ singular? - [ ] Spoiler: $X_\sigma \cong V(y^2-xz)$ which is singular at the origin. - [ ] Descibe $X_\sigma$ as a quotient. - [ ] Spoiler: $X_\sigma \cong {\mathbb{C}}^2/{\mathbb{Z}}^2$ where the action is $(-1)\cdot {\left[ {x,y} \right]} \coloneqq{\left[ {-x, -y} \right]}$ and the invariants are $\left\langle{x^2, xy, y^2}\right\rangle$ - [ ] Show that $\sigma={ \mathrm{Cone} }\left(e_{1}, e_{2}, e_{3}, e_{1}-e_{2}+e_{3}\right) \subseteq \mathbb{R}^{3}$ is not simplicial. - [ ] Draw $\sigma$ - [ ] Spoiler: ![](attachments/Pasted%20image%2020220605141655.png) - [ ] Find $S_\sigma$, ${\mathbb{C}}[S_\sigma]$, and $X_\sigma$. Is $X_\sigma$ smooth, singular, an orbifold..? - 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). - [ ] Describe the faces in the following convex polytope: ![](attachments/Pasted%20image%2020220605142615.png) - [ ] Spoiler: one 0-dimensional (point), three 1-dimensional (rays), three 2-dimension (cones). - [ ] Hirzebruch surfaces $H_r$. - [ ] Show that $H_r$ is a ruled toric variety. - [ ] Describe $H_r$ as a vector bundle. - [ ] Spoiler: $H_r$ is the bundle over ${\mathbb{P}}^1$ associated to the sheaf ${\mathcal{O}}(0) \oplus {\mathcal{O}}(-r)$. - [ ] Describe $H_r$ as $X_\Sigma$ for a fan. - Spoiler: ![](attachments/Pasted%20image%2020220605144233.png) - [ ] Show that $H_0 = {\mathbb{P}}^1 \times {\mathbb{P}}^1$. - [ ] Show that $H_1 = \operatorname{Bl}_0({\mathbb{P}}^2)$. - [ ] Show that $H_r$ is smooth iff $r\leq 1$. - [ ] Show that for $r\geq 2$, $H_r = \operatorname{Bl}_p {\mathbb{P}}(1,1,r)$ for $p$ a singular point. - [ ] What is the fan for $X = \operatorname{Bl}_0({\mathbb{C}}^2)$? Use this to compute $\operatorname{Div}(\chi^{u_i})$ for the generators $u_i$ and to show $Cl(X) = {\mathbb{Z}}$ with generators $[D_1] = [D_2] = -[D_0]$. - 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 ![](attachments/Pasted%20image%2020220605164554.png) - [ ] Use rays and the exact sequence for the class group to show ${ \operatorname{Cl}} ({\mathbb{P}}^n) = {\mathbb{Z}}$. - [ ] Spoiler: $m \mapsto {\left[ {-\sum m_i, m_1,\cdots, m_n} \right]}$ and ${\left[ {a_0,\cdots, a_n} \right]} \mapsto \sum a_i$, using that $\sum e_i = 0$ is the only relation among the primitive generators. - [ ] Compute the class group of $\Sigma=\left\{\sigma=\operatorname{Cone}\left(e_{2}, 2 e_{1}-e_{2}\right), \text { Cone }\left(e_{2}\right), \text { Cone }\left(2 e_{1}-e_{2}\right), 0\right\}$ ![](attachments/Pasted%20image%2020220605170749.png) - [ ] Spoiler: `

`{=html} > [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMCwwLCIwIl0sWzIsMCwiXFxaWl4yIl0sWzQsMCwiXFxaWl4yIl0sWzYsMCwiXFxaWi8yXFxaWiJdLFs4LDAsIjAiXSxbMiwxLCJtIl0sWzQsMSwiKG1fMiwgMm1fMSAtIG1fMikiXSxbNCwyLCIoYV8xLCBhXzIpIl0sWzYsMiwiYV8xICsgYV8yIFxcbW9kIDIiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XSxbNSw2LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzcsOCwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==) - [ ] Work out the following example for $X_\Sigma = {\mathbb{P}}^1\times {\mathbb{P}}^1$: ![](attachments/Pasted%20image%2020220605191828.png) - [ ] Describe the Segre embeddeding ${\mathbb{P}}^1 \times {\mathbb{P}}^1 \to {\mathbb{P}}^3$ and its corresponding polytope - [ ] Spoiler: ${\left[ {a:b} \right]} \times {\left[ {c:d} \right]} \mapsto {\left[ {ac: ad: bc: bd} \right]}$, corresponding to $V = {\left[ {v_1, v_2, v_3, v_4} \right]}^t$ where $v_1 = {\left[ {1,0,1,0} \right]}, v_2 = {\left[ {1,0,0,1} \right]}, v_3 = {\left[ {0,1,1,0} \right]}, v_4 = {\left[ {0,1,0,1} \right]}$, which is a cone on a square. ## 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 $k[{\mathbb{N}}]$. ## Unsorted - [ ] How can a toric variety be realized as a GIT quotient? - [ ] Spoiler: $X_\Sigma = \dcoset{{\mathbb{C}}^r}{Z}{{ \mathbin{/\mkern-6mu/}}G}$ where $r$ is the number of one-dimensional rays. Here $G = \mathop{\mathrm{Hom}}({ \operatorname{Cl}} (X_\Sigma), {\mathbb{C}}^{\times})$ fits into a SES $1\to G\to ({\mathbb{C}}^{\times})^{{\sharp}\Sigma(1)} \to T_N \to 1$ obtain by applying $\mathop{\mathrm{Hom}}({-}, ({\mathbb{C}}^{\times})^n)$ to $1\to M \to ZZ^{{\sharp}\Sigma(1)} \to { \operatorname{Cl}} (X_\Sigma) \to 1$ when $\Sigma(1)$ spans ${\mathbb{R}}^n$. - [ ] How is $Z$ defined in the above quotient? - [ ] Spoiler: write the coordinate ring of ${\mathbb{C}}^{{\sharp}\Sigma(1)}$ as ${\mathbb{C}} { \left[ \scriptstyle {\left\{{x_\rho {~\mathrel{\Big\vert}~}\rho \in \Sigma(1)}\right\}} \right] } $, define monomials $x^\sigma \coloneqq\prod_{\rho \not\in \sigma(1)} x_\rho$, take the monomial ideal $C \coloneqq\left\langle{x^\sigma {~\mathrel{\Big\vert}~}\sigma\in \Sigma_{\max}}\right\rangle$, and define $Z \coloneqq V(C)$, a union of coordinate planes of codimension 2 or more. - [ ] What is a primitive collection? - [ ] Characterize when $X_\Sigma$ is a good geometric quotient. - [ ] Spoiler: when $\Sigma$ is simplicial. - [ ] Describe how to see the orbits in a toric variety. - [ ] Spoiler: orbits are indexed by cones. - [ ] Describe the orbit of $T$ on $X$. - [ ] Spoiler: all coordinates nonzero. - [ ] What is the algebraic moment map $X\to P$ for $P$ a polytope?