--- sort: 060 title: Toric Varieties flashcard: "Orals::Toric Varieties" --- # Toric Varieties - [x] What is the **character lattice** $M$? What is its **dual lattice** $N$? #syllabus ✅ 2022-12-07 - [ ] $M =$ **characters**, $N =$**cocharacters/1PSs**. - [ ] Set $T^n \da \GG_m^n$, then $M \da \Hom(T^n , \GG_m)$ is the character lattice. Contains elements $\chi_m$ where $\vector t \mapsto \prod t_i^{m_i}$. Then $N \da \Hom(\GG_m, T^n)$ is the lattice of cocharacters/1-parameter subgroups, contains elements $\lambda^u$ where $t\mapsto \vector t^u \da (t^{u_i})$. Use the perfect pairing $\inp m n \da \ell$ where $\ell$ is the unique integer such that $\chi_m \circ \lambda^n: \GG_m\to \GG_m$ is $t\mapsto t^\ell$. Note $N\tensor_\ZZ \GG_m \cong T$. - [x] What is a **strongly convex rational polyhedral cone**? #syllabus ✅ 2022-12-07 - [ ] **Cone**: $\RR_{\geq 0} \sigma \subseteq \sigma$. - [ ] **Strongly convex**: $\sigma \intersect (-\sigma) = 0$, or contains no nonzero subspace of $N_\RR$, or $\dim \sigma\dual = \dim M$. - [ ] **Rational**: generated by lattice elements $\ts{\vector v_i} \subseteq N$. - [ ] **Polyhedral**: finitely generated. - [x] What is the **dual of a cone**? #syllabus ✅ 2022-12-07 - [ ] $\sigma\dual \da \ts{m\in M \st \inp n m \geq 0 \,\, \forall n\in \sigma}$. - [x] Compute the toric variety associated to $\Cone(v_1, v_2)$ where $v_1 = (1, 4)$ and $v_2 = (1, 4)$. ✅ 2022-12-07 - [ ] Can get perps by $\pi$ rotation $$e_1 \mapsto e_2, e_2 \mapsto-e_1 \rightsquigarrow\left[\begin{array}{cc} 0 & -1 \\1 & 0\end{array}\right]$$ - [ ] So $v_1^\perp = (-1, 2), v_2^\perp = (-4, 1)$. Flip $v_2^\perp$ since normal was oriented outside of cone to get $(4, -1)$, take cone spanned: - [ ] ![](attachments/Pasted%20image%2020221207225009.png) - [ ] Find basis $(4, -1), (1,0), (0, 1), (-1, 2)$ corresponding to $u,v,w,t \da x^4 y\inv, x, y, x\inv y^2$. - [ ] Know this should be 1-dimensional, so find 3 relations to get $$X_\sigma = \spec k[x^4 y\inv, x, y, x\inv y^2] = \spec { k[u,v,w,t] \over (uv-w^2, wt-v^4, ut-v^3w) }$$ - [x] Discuss how to construct an affine toric variety from a fan. #syllabus ✅ 2022-12-07 - [ ] $\sigma \leadsto S_\sigma \da \sigma\dual \intersect M \leadsto X_\sigma \da \spec \CC[S_\sigma]$ where $\CC [S_\sigma] \da \ts{ \sum_{s\in S_{\sigma}} c_s \chi^s \st c_s\in \CC \text{ and }c_s = 0 \ae}$ with character multiplication. - [x] What is a **fan**? #syllabus ✅ 2022-12-07 - [ ] A collection of SCRPCs closed under taking faces and the intersection of any two cones is a face, where $\tau \leq \sigma$ is a face iff $\tau = H_m \intersect \sigma$ for some hyperplane corresponding to $m\in \sigma\dual \subseteq M_\RR$ and $H_m \da \ts{n\in N_\RR \st \inp m u = 0} \subseteq N_\RR$. - [x] What is a **convex set**? #syllabus ✅ 2022-12-07 - [ ] For all $p,q\in S$, the line $tp + (1-t)q\in S$. In dimension 2, equivalently for any $\ts{p_i} \subset S$ and any scalars $\ts{\lambda_i}$ with $\sum \lambda_i = 1$, $p \da \sum \lambda_i p_i\in S$. - [x] What is the toric variety associated to a polytope? #syllabus ✅ 2022-12-07 - [ ] For $P\subseteq M$, take the cone over $P\times\ts{1} \subseteq M_\RR \times \RR$ to get a fan whose cones are cones over proper faces of $P$ - [ ] Note $F\leq P$ is proper iff $F = P \intersect(H_n + r))$ for some $r$, i.e. equal to intersection with some affine hyperplane so $F = \ts{p\in P \st \inp n p = r}$.) - [ ] Alternatively: take the normal fan of $P$ in $N$ to define $X_P$ - [ ] Alternatively: $X_P \da \Union_{m\in M\intersect P} \spec \CC[\sigma_{\hat m} \intersect M]$ where $\sigma_{\hat m}\da \Cone(P\intersect M-m)$. - [ ] How these are related: normal fan for $P \cong$ proper faces over $P^\circ$. - [x] What is the polytope $Q_D$ associated to a Weil divisor $D$ on $X_P$? ✅ 2022-12-07 - [ ] Write $D = \sum a_\rho D_\rho$, then $$Q_D = \ts{m\in M_\RR \st \inp m \rho \geq -a_\rho \, \forall \rho\in \Sigma(1)}.$$ For the anticanonical one gets $Q_{-K_{X_p}} = P$. - [x] Discuss the **moment map** for $X_\Sigma$. ✅ 2022-12-07 - [ ] $\mu_{\Sigma}: \CC^{\size \Sigma(1)} \to \RR^{r-n}$ for $n = \dim N_\RR$. - [x] Discuss the **orbit-cone correspondence**. #syllabus ✅ 2022-12-07 - [ ] Each cone $\sigma$ determines a distinguished point $x_\sigma \in U_\sigma$ corresponding to the map $S_\sigma \da \sigma\dual \intersect M \to \ts{0, 1}$ where $m\mapsto 1$ if $m\in \sigma^\perp$ and 0 otherwise where $\sigma^\perp \da \ts{m\in M_\RR \st \inp n m = 0\,\, \forall n\in \sigma}$. Then there is a bijection from $\Sigma$ to $T\dash$orbits in $X_\Sigma$ where $\sigma \mapsto T\cdot x_\sigma$ and an $r\dash$dimensional cone corresponds to an $n-r\dash$dimensional orbit, e.g. $\dim \sigma =0 \mapstofrom \GG_m^n$ and $\dim \sigma = n \mapstofrom T\dash$invariant points, and the face poset corresponds to the orbit closure poset (contravariantly). - [x] Discuss as many examples of fans and their corresponding toric varieties as you can. #syllabus ✅ 2022-12-08 - [ ] $(\CC\units)^n, \CC^n, \PP^n, \FF_n$ for any $n$ - [ ] The rational normal cone $\hat C_d$ gotten from $(d,-1), (0,1)$. - [ ] The cone over a cone $V(y^2-xz)$, gotten from $(0, 1), (2, -1)$: ![](attachments/Pasted%20image%2020221208002529.png) - [ ] $\Bl_m X_\Sigma$ for any $\Sigma$. - [ ] $V(xy-zw)$ gotten from $(1,0,0), (0, 1,0), (1,0,1), (0,1,1)$. - [ ] Any products of the above. - [x] Show that a divisor class group may not be torsionfree. ✅ 2022-12-08 - [ ] $\Cl(\hat C_d) = \ZZ/d\ZZ$, see 4.14 Cox. - [x] What is a toric morphism? #syllabus ✅ 2022-12-08 - [ ] On lattices: images of cones are contained in cones. A map $\phi: N_1\to N_1$ where for each $\sigma_1 \subseteq N_{1, \RR}$ there is a $\sigma_2 \subseteq N_{2, \RR}$ with $\phi(\sigma_1) \subseteq \sigma_2$. At the level of varieties: $\phi: X_{\Sigma_1} \to X_{\Sigma_2}$ a morphism of varieties with $\phi(T_{1})\subseteq T_2$ which restricts to a group morphism. At the level of abstract toric varieties: for $V_i = \spec \CC[S_i]$ and $\phi: V_1\to V_2$, toric iff $\phi^*: \CC[S_2] \to \CC[S_1]$ is induced by a semigroup morphism $(\phi^*)_*: S_1\to S_2$. - [x] When is a toric variety smooth? #syllabus ✅ 2022-12-08 - [ ] Iff each affine patch is smooth, and $U_\sigma$ is smooth iff admits generators which can be extended to a basis of $N$. In this case $U_\sigma \cong \AA^k \times \GG_m^{n-k}$ where $k = \dim \sigma$. - [x] When is a toric variety proper/complete? #syllabus ✅ 2022-12-08 - [ ] When $\abs{\Sigma} = N_\RR$, i.e. full support. Why: each $n\in N_\RR$ determines a one-parameter subgroup $\lambda^n$, and if $n=(n_1, n_2)$ then one looks at $\lim_{t\to 0}\lambda^n(t)$ -- if $n$ is in a cone, the limit exists, else not. E.g. for $\PP^2$ the torus is $(1,s,t)$ and $\lambda^n(t) = (1, s^{n_1}, t^{n_2})$ and one gets: ![](attachments/Pasted%20image%2020221208004055.png) and one gets the distinguished limit points ![](attachments/Pasted%20image%2020221208004152.png) - [x] How is a line bundle on $X$ described? #syllabus ✅ 2022-12-08 - [ ] Every $\mcl\in \Pic(X_\Sigma)$ must be trivial when restricted to $T^n$ so $\mcl = \OO_X(D)$ for some $D\in \Cart(X_\Sigma)$. - [x] Give a toric example of a Weil divisor that is not Cartier. #syllabus ✅ 2022-12-09 - [ ] Take Cone $\left(e_2, d e_1-e_2\right)$, then check $M \rightarrow \operatorname{Div}_T\left(U_\sigma\right)$ by $\left[\begin{array}{cc}d & -1 \\ 0 & 1\end{array}\right]$ so $\mathrm{Cl}\left(U_\sigma\right)=\mathbf{Z} / d \mathbf{Z}($ $0 = \operatorname{dim} \operatorname{div}\left(\chi^{e_1}\right)=d D_1$ and $\left.0 =\operatorname{dim} \operatorname{div}\left(\chi^{e_2}\right)=-D_1+D_2\right)$ but $U_\sigma$ is affine so $\operatorname{Pic}\left(U_\sigma\right)=0$. - [ ] Not commutativity of $\Div \to \Cl$ and $\CDiv\to \Pic$ forces $D \da D_1$ to be Weil but not Cartier for any $d\geq 1$. - [ ] In fact $\Pic(X) = \gens{mD_2}_\ZZ$ and $A_1(X) = \gens{D_2}_\ZZ$. - [x] What is $\rank \Pic(X_\Sigma)$? #syllabus ✅ 2022-12-08 - [ ] If $\Sigma$ is simplicial, always $\size\Sigma(1)-\dim \NN_\RR$. E.g. $\Pic(\FF_n) = 4-2 = 2$, generated by a special curve and a fiber. - [x] When is $D$ **globally generated/bpf**? ✅ 2022-12-08 - [ ] Polytope side: $\OO(D)$ is globally generated iff $P_D$ is the convex hull of $\ts{u_\sigma \st \sigma \in \Sigma(n)}$. - [ ] $\abs{D}$ is bpf iff $\psi_D$ is convex - [ ] $\psi_D$ is **convex** if $\psi(tv + (1-t)w)\geq t\psi(v) + (1-t)\psi(w)$ - [x] When is $D$ **ample/bpf**? ✅ 2022-12-08 - [ ] $\OO(D)$ is ample iff $u_{\sigma}\neq u_{\sigma'}$ for distinct $\sigma\neq \sigma'\in \Sigma(n)$ and $P_D$ is $n\dash$dimensional with vertices $u_\sigma$. - [x] When is $D$ **very ample**? #syllabus #important ✅ 2022-12-09 - [ ] $D$ is very ample iff $\psi_D$ is strictly convex **and** $S_\sigma$ is generated by $\ts{u - u(\sigma) \st u\in P_D\intersect M}$ where $u(\sigma)$ is the linear function associated to $\sigma$ defined by $\psi_D(\wait) = \inp{u(\sigma)}{\wait}$ so $D = \sum - \psi_D(\rho_i) D_{\rho_i}$ since $\psi_D(\rho_i) = a_i$ in $D = \sum a_i D_{\rho_i}$. - [ ] $\psi_D$ is **strictly convex** if convex and $u(\sigma), u(\sigma')$ are not equal for distinct maximal cones $\sigma,\sigma'$. - [ ] If $X$ is smooth complete, ample iff very ample. - [ ] Otherwise check if $P_D$ is an ample polytope: for every vertex $v$, the semigroup $S_{P, v} \da \NN(P_v)$ where $P_m \da P\intersect M-v \da \ts{p-v \st p \in P \intersect M}$ is **saturated** in $M$. - [x] Discuss $\Pic(X_\Sigma)$ and $\Cl(X_\Sigma)$. ✅ 2022-12-08 - [ ] The fundamental SESs: ![](attachments/Pasted%20image%2020221208004842.png) - [x] Discuss **global sections** of line bundles. #syllabus ✅ 2022-12-08 - [ ] $H^0(\OO(D)) = \bigoplus_{m\in P_D \intersect M} \CC\cdot \chi^m$. - [x] Classify complete smooth **toric surfaces**. #syllabus ✅ 2022-12-08 - [ ] Given by sequences $\ts{v_0,\cdots, v_{d-1}, v_d = v_0}$ in counterclockwise order such that $\ts{v_i, v_{i+1\mod d}}$ generate $N$. - [ ] Yields $a_i v_i = v_{i-1} + v_{i+1}$. - [ ] For $d=3$, must be $\PP^2$, and $d=4$ yields $\FF_n$. For $d\geq 5$, always a blowup of $\PP^2$ or $\FF_n$ at $T\dash$fixed points at $\tilde v_i \da v_{i} + v_{i+1}$. - [ ] The integers $a_i$ must satisfy $\prod\left[\begin{array}{cc}0 & -1 \\ 1 & a_i\end{array}\right]= \id$ and $\sum a_i=3 d-12$. - [x] What is a **Gorenstein Fano variety**? ✅ 2022-11-29 - [ ] For $X$ complete and normal: $-K_X$ is Cartier and ample. - [x] Classify smooth **Fano** toric surfaces. ✅ 2022-11-29 - [ ] Fano polytopes: $0\in \mathrm{int} P$ and the vertices of every facet form a lattice basis. $P$ is reflexive, so $P\dual$ is a lattice polytope and $X_{P\dual}$ is smooth toric Fano whose normal fan comes from taking cones over proper faces of $P$ -- every smooth toric surface arises this way. Toric *Gorenstein* Fanos: just 16 equivalence classes. If $X_\Sigma$ is projective Gorenstein Fano, then $P_{-K_{X_\Sigma}}$ is reflexive, and every $X_P$ for $P$ reflexive is Gorenstein Fano. - [x] Classify toric **del Pezzo** surfaces. #syllabus ✅ 2022-12-09 - [ ] Smooth projective Fanos of dimension 2, correspond to smooth reflexive polytopes. - [ ] There are exactly 5: $\PP^1\times \PP^1, \Bl_i \PP^2$ for $i=0,1,2,3$. Get the last three by blowing up (successively) the 3 torus-invariant points on $\PP^2$. ![](attachments/Pasted%20image%2020221201023435.png) - [x] What is a **smooth polytope**? ✅ 2022-12-01 - [ ] For $\dim M = n$, each vertex meets exactly $n$ edges and the primitive vectors along each edge generate a lattice basis. - [x] How does one **blowup** a toric variety? #syllabus ✅ 2022-12-08 - [ ] Subdivide any cone. - [x] How does one find the **minimal resolution** of a toric variety? #syllabus ✅ 2022-12-08 - [ ] Take the convex hull of $\sigma \intersect N$ and blowup along those lattice points: ![](attachments/Pasted%20image%2020221208012620.png) - [x] What are the **Weil divisors** of a toric variety? #syllabus ✅ 2022-12-08 - [ ] $\Div_T(X_\Sigma) \cong \ZZ^{\Sigma(1)}$, i.e. each $\rho\in \Sigma(1)$ corresponds to $D_\rho$. A general Weil divisor is $D = \sum_{\rho \in \Sigma(1)} a_\rho D_\rho$. - [x] What are the **Cartier divisors** on a toric variety? #syllabus ✅ 2022-12-08 - [ ] $\CDiv_T(U_\sigma) \cong M/M(\sigma)$ where $M(\sigma) \da \sigma\dual \intersect M$, and $\CDiv_T(X_\Sigma) = \cocolim_{\sigma \in \Sigma} M/M(\sigma)$. More concretely described by Cartier data $\ts{(U_\sigma, m_\sigma)}$ where $\ro{D}{U_\sigma} = \div(\chi^{m_\sigma})$ for $m_\sigma\in M$ with $\inp \rho {m_\sigma} = -a_\rho$ for every $\rho\in \Sigma(1)$, where $D = \sum a_\rho D_\rho$. - [ ] Can also obtain ffrom polytopes: write $P = \ts{m\in M_\RR \st \inp {\mu_F} m \geq -a_F \,\,\text{ for all facets } F\leq P}$ where $\mu_F$ is the minimal generator along the normal to the facet $F$. Then take $D \da \sum a_F D_F$ where $D_F$ is the divisor for the ray associated to $F$ in $X_P$. - [x] What is $K_X$ for $X$ toric? #syllabus ✅ 2022-12-08 - [ ] $\omega_X \cong \OO(-\sum_\rho D_\rho)$, so $K_X \sim -\sum_\rho D_\rho$. As a consequence, for $X \da \PP^n$ we have $\omega_X = \OO(-n-1)$ since there are $n+1$ generators in the fan and all $D_i\sim D_j$. - [x] When is a toric variety **Fano**? #syllabus ✅ 2022-12-09 - [ ] Need $-K_X$ Cartier and ample, so $\OO(\sum_\rho D_\rho)$ ample. - [x] Find the **polytope of a divisor**. ✅ 2022-12-08 - [ ] For $\PP^2$, it's $3\Delta$. ![](attachments/Pasted%20image%2020221208015307.png) ![](attachments/Pasted%20image%2020221208015324.png) - [x] What is $\pi_1 X$ of a toric variety? #syllabus ✅ 2022-12-08 - [ ] $\pi_1 X_{\Sigma} \cong N/N'$ where $N' \da \gens{\sigma \intersect N \st \sigma\in \Sigma}$. - [x] What is $\chi(X)$ for a toric variety? #syllabus ✅ 2022-12-08 - [ ] $\chi(X_\Sigma) = \size \Sigma(n)$. - [x] How are **intersection numbers** of divisors computed? #syllabus ✅ 2022-12-08 - [ ] For smooth complete toric surfaces: write $\rho_{i-1} + \rho_{i+1} = b_i \rho_i$ as a linear dependence, then $$D_i \cdot D_j=\left\{\begin{array}{lll}0 & \text { if } & |i-j|>1 \\1 & \text { if } & |i-j|=1, \\-b_i & \text { if } & i=j\end{array}\right.$$ - [x] Example: ✅ 2022-12-08 ![](attachments/Pasted%20image%2020221208020507.png) - [x] What is the **quadric cone**? #syllabus ✅ 2022-12-08 - [ ] $V(z^2-xy)$. - [x] What is a **reflexive polytope**? ✅ 2022-11-29 - [ ] One with facet presentation $$P = \ts{m\in M_\RR \st \inp m {\mu_F} \geq -1 \text{ for all facets } F}$$ - [ ] The origin is contained in its interior and every facet is at lattice distance 1 from the origin (guarantees that the polar dual is again a lattice polytope). - [x] What is the **polar dual** of a polytope? #syllabus ✅ 2022-11-29 - [ ] For reflexive polytopes, since $a_F = 1$ for all $F$, $$P^\circ = \mathrm{Conv}(\mu_F \mid F \text{ is a facet of } P)$$ - [ ] $$P^\circ \da \ts{n\in N \st \inp n p \geq -1\, \forall p\in P}$$ - [x] What is $P_{-K_X}$ for $X = \PP^2$? What is its dual? ✅ 2022-11-29 - [ ] ![](attachments/Pasted%20image%2020221129103949.png) - [x] How can you read the projective embedding off of a polytope? ✅ 2022-11-29 - [ ] ![](attachments/Pasted%20image%2020221129104521.png) - [x] What is an **integral** support function? ✅ 2022-11-29 - [ ] $\phi: \abs{\Sigma} \to \RR$, linear on each cone, and integral if $\phi(\abs{\Sigma} \intersect N) \subseteq \ZZ$. - [x] What are the global sections of $\OO(D)$ for $D\in \Div_T(X)$? ✅ 2022-11-29 - [ ] $\Gamma\left(X_{\Sigma}, \mathscr{O}_{X_{\Sigma}}(D)\right)=\bigoplus_{m \in P_D \cap M} \mathbb{C} \cdot \chi^m,$ - [x] When is $D$ **bpf**? ✅ 2022-11-29 - [ ] Supposing the maximal dimensional cone is $n$, $D$ is bpf iff $\OO(D)$ is globally generated iff $m_\sigma\in P_D$ for all $\sigma\in \Sigma(n)$. - [x] What is the **support function** associated to $D$? ✅ 2022-11-29 - [ ] For $D=\sum a_p D_p$, uniquely determined by: $\phi_D$ is linear on each cone, $\phi_D(u_p) = -a_p$ where $u_p$ are primitive ray generators. - [x] What is a **convex** support function? ✅ 2022-11-29 - [ ] $\varphi(t u+(1-t) v) \geq t \varphi(u)+(1-t) \varphi(v)$. - [x] What is a **strictly convex** support function? ✅ 2022-11-29 - [ ] For every $\sigma\in \Sigma(n)$, $\inp{m_\sigma}{u} = \phi_D(u) \iff u\in \sigma$. - [x] When is $D$ **ample**? ✅ 2022-11-29 - [ ] Iff $\phi_D$ is strictly convex. - [x] Show that $\PP^2$ is Fano. ✅ 2022-12-08 - [ ] $\omega_X = \OO(-\sum_\rho D_\rho) = \OO(-D_1 - D_2\cdots - D_3) = \OO(-3)$ so $-\omega_X = \OO(3)$ which is ample. - [x] Why are toric varieties **normal**? ✅ 2022-12-08 - [ ] Need local rings to be integrally closed; the local ring at $x_\sigma$ is $R_\sigma = \intersect _{?} R_{\tau_i}$, but each $R_{\tau_i}$ is of the form $\CC[x_1,x_2^{\pm 1},\cdots, x_n^{\pm 1}]$ and intersections of integrally closed domains are again integrally closed.. - [x] What is the polytope associated to a Cartier divisor? ✅ 2022-12-09 - [ ] $D = \sum a_i D_i\implies P_D = \ts{u\in M_\RR\st \inp {u} {\rho_i} \geq a_i}$. - [x] Classify smooth **affine** toric varieties. ✅ 2022-12-08 - [ ] Always of the form $X_\sigma = \AA^k \times \GG_m^{n-k}$. - [x] Classify **smooth complete toric surfaces**. ✅ 2022-12-08 - [ ] On $d=3$ lattice points: $\PP^2$. - [ ] On $d=4$ lattice points: $\FF_n$ a Hirzebruch surface $(\FF_0 \cong \PP^1\times \PP^1)$ - [ ] On $d\geq 5$ lattice points: $\Bl_m X$ for $X = \PP^2, \FF_n$ for some $n, m$. In general one has $$\left(\begin{array}{c}v_i \\ v_{i+1}\end{array}\right)=\left(\begin{array}{cc}0 & 1 \\ -1 & a_i \end{array}\right)\left(\begin{array}{c} v_{i-1} \\ v_i\end{array}\right), \quad \prod_i\left[\begin{array}{cc}0 & 1 \\-1 & a_i\end{array}\right]=\mathrm{id}, \quad \sum_i a_i=3 d-12$$ where $d$ is the number of rays. - [x] How do blowups and contractions of fans change self-intersection numbers? ✅ 2022-12-08 - [ ] ![](attachments/Pasted%20image%2020221201163739.png) - [x] How does one construct weighted projective space as a toric variety? ✅ 2022-12-11 - [ ] $\PP(q_0,\cdots, q_n) \leadsto \Conv(0, k_1e_1,\cdots, k_ne_n) - (1,1,\cdots,1)$ where $k_i \da \qty{q_i\over \sum q_i}\inv\in \ZZ$. - [ ] Discuss how to compute $\Pic(X)$ and $\Cl(X)$ for a toric variety. #syllabus - [ ] What is the rational cuspidal curve? #syllabus