--- created: 2024-05-05T11:42 updated: 2024-05-05T11:42 --- - Show that $\sigma$ is a simplicial cone if $\sigma\dual$ is of maximal dimension - Classify smooth toric varieties in all dimensions - See Oda's Convex Bodies book. - Is there a clear algorithm for finding generators of $S_\sigma$? - In some cases, it's more or less clear what to do: ![[Pasted image 20230825115934.png]] and ![[Pasted image 20230825115942.png]] - Finding dual cones: - Given $\sigma = \Cone(v_1, v_2)$, let $M$ be the matrix with $v_i$ as columns; then the columns of $M^{-t}$ are the primitive rays of $\sigma\dual$? - E.g. $v_1 = (0, 1)$ and $v_2 = (2, -1)$ yields $\sigma \dual$ generated by $(1, 2)$ and $(1, 0)$ and this checks out. Note sure if it generalizes. ![[Pasted image 20230825115623.png]] - Some interesting toric varieties: - $\sigma = \Cone(e_1, e_2, e_3, e_1+e_2 - e_3) \subseteq \ZZ^3$ yields $\CC\adjoin{x,y,z,xyz\inv} = \CC[x,y,z,w]/\gens{xy-zw}$. - $\sigma = \Cone(e_1, e_2, e_1 + e_3, e_2 + e_3)$. - ![[Pasted image 20230825115735.png]] - Faces of a cone $\sigma \subseteq N$: subsets of the form $\tau = \sigma \intersect m^\perp$ for some $m\in \sigma\dual \intersect M$. - $\Cone(S) = \RR_{\geq 0} S = \ts{\vector v\da \sum \lambda_i s_i \st \lambda_i \in \RR_{\geq 0}}$. - $srm{Conv}(S) = \ts{\vector v\in \Cone(S) \st \sum \lambda_i = 1}$. - Strongly convex: $\sigma \intersect(-\sigma) = \ts{0}$. - Example: $Q1$ - Non-example: $Q1 \union Q2$, since $\sigma \intersect (-\sigma) = \RR$. - $y\in \sigma\dual \iff \ts{z\in \inp y z \geq 0} \contains \sigma$. - Show $\sigma \da \Cone(e_2, 2e_1 - e_2)$ yields $\CC[x,y,z]/\gens{y^2=xz}$. --- - A Fano curve is rational, i.e. $\PP^1$. - Toric del Pezzos: $\PP^1\times \PP^1$ or $\Bl_k \PP^2$ for $k\leq 8$. - Unirational: dominated by a rational variety, where a dominant morphism is $\pi: X\torational Y$ with some $U \subseteq X$ open such that $\pi(U)$ dense. - Gorenstein: $K_X \in \CDiv(X)$ is Cartier. Equivalently, $\omega_X = \OO_X(K_X)$ is a line bundle, i.e. in $\Pic(X)$. - Reflexive polytopes: $P$ is reflexive iff the polar dual $P^\circ$ is again a lattice polytope. - Example: the triangle spanned by $(0, 1), (1, 0), (-1, -1)$ - Non-example: the triangle spanned by $(0, 2), (1, 0), (-1, -1)$. Note that the interior of this polytope contains a lattice point other than the origin. - ![[Pasted image 20230825152319.png]] - Equivalently, $P$ has a presentation of the form $P = \ts{m\in M_\RR \st \inp {m}{\mu_F} \geq -1 \forall \text{ facets }F \text{ of } P}$. - What is $\mu_F$? Inward pointing unit normal of face $F$. - Can present faces of a polytope $P \subseteq M$ as $F = \ts{p\in P \st \inp {\mu_F}{p} \geq -1 \, \forall p\in P}$ - Can then present the polar dual as $P^\circ = \ts{u\in N \st \inp u p \geq -1 \, \forall p\in P}$. - There is a duality $$ \mathrm{Faces}^i(P) \mapstofrom \mathrm{Faces}^{n-1-i}(P^\circ) \qquad F \mapsto F^\circ $$ where $\dim F + \dim F^\circ = \dim M - 1$. - If $P$ is reflexive, then $0$ is the only lattice point in $\mathrm{Interior}(P)$, and we can write $P^\circ = \mathrm{Conv}\qty{\ts{ \mu_F \mid F \in \mathrm{Facets}(P) }}$. - In this case $P^\circ$ is a lattice polytope and is reflexive. - Facet presentation for $P$: $P = \ts{m\in M_\RR \st \inp m {\mu_F} \geq a_F \, \forall F}$. - E.g. consider ![[2023-08-01 Old Notes 2023-08-25 13.21.59.excalidraw]] Here $\mu_{F_i} = e_i$ and $e_{-i} = -e_i$ for $i=1, 2$, where the sides are $F_{-1}, F_{-2}, F_1, F_2$ in counterclockwise order starting from the east side of the square, which yield corresponding lattice points $f_1, f_2, f_{-1}, f_{-2}$ in the same order. - Constructing the facet polytope: - For $P \subseteq M_\RR$, construct $\Delta_{\text{Facet}}(P) \subseteq N_\RR$. - For every face $F$ of $P$, set $\sigma_F \da \ts{n\in N_\RR \st \inp f n \leq \inp p n \,\, \forall f\in F, p\in P} \subseteq N_\RR$. - One can realize generators of $\sigma_F\dual$ as $\ts{p-f \st p\in P, f\in F} \subseteq M_\RR$, which point from the face into the other vertices of $P$; ![[2023-08-01 Old Notes 2023-08-25 13.33.48.excalidraw]] - One can check that $\ts{\sigma_F}$ form a fan $\Delta(P)$; Fulton shows that if $0\in P$ ten $\Delta(P)$ can be realized as cones over proper faces of $P^\circ$. - If $0\in P$ then $\Delta(P)$ is complete. - This defines a toric variety $X_P$. - $X_P$ is Fano when for every facet $F$ of $P$, the vertices of $F$ form a $\ZZ\dash$basis for $M_\RR$.[^1] - A trick for checking this: $\ts{m_1, m_2}$ form a $\ZZ\dash$basis iff $\mathrm{Conv}(0, m_1, m_2) \intersect M = \ts{0, m_1, m_2}$. - Given $P = \ts{\inp m {\mu_F} \geq - a_F}$, then $D_P \da \sum a_F D_F$ is always Cartier and ample. - If $P$ is reflexive then $X_P$ is always Gorenstein Fano. - There are correspondences: - Gorenstein toric Fano varieties $\mapstofrom$ Reflexive polytopes - Smooth Fano toric surfaces $\mapstofrom$ Reflexive 2-dimensional polytopes - An example of a polytope that does not define a smooth variety: ![[Pasted image 20230825152557.png]] - When is $X_P$ smooth, in terms of the polytope $P$? Seems to be if any face $F$ contains a non-vertex lattice point. - There are 5 smooth 2d reflexive polytopes: ![[Pasted image 20230825152728.png]] - Some examples: - Take the 4-simplex $\Delta_4 \da \mathrm{Conv}(0, e_1, e_2, e_3, e_4)$ and set $P \da 5\Delta_4 - \sum e_i$. Then $P^\circ = \mathrm{Conv}(e_1, e_2, e_3, e_4, -\sum e_i)$, which contains a quintic threefold. - Take $\Delta_2 \da \mathrm{Conv}(0, e_1, e_2)$ and $P \da 3\Delta_2 - (e_1 + e_2)$, then $P^\circ = \mathrm{Conv}(e_1, e_2, -e_1-e_2)$: ![[Pasted image 20230825153453.png]] - Not all fans come from convex polytopes. - Theorem (Cox 8.3.4): $X_\Sigma$ is a normal toric Gorenstein Fano variety $\iff P_{(X_\Sigma, -K_{X_\Sigma})}$ is a reflexive polytope. - $\impliedby: a_F = 1 \implies D_P$ ample $\implies -K_X$ ample. - $\implies$: Use presentation $P = \ts{m\st \inp{u_\rho}m \geq -1 \, \forall\rho\in \Sigma(1) }$. - What is the Mori program? - An example of when a divisor is ample in $\PP^n$: - Let $X = D\in \Div(\PP^n)$, then $K_D = (K_{\PP^n} + D)\mid_D = (-(n+1)H + \deg(D) H)\mid_D = (\deg(D) - n - 1)\mid_D = (\deg(D) - (n+1)H)\mid_D \da mH\mid_D$ and $mH$ is ample iff $m > 0$, so $-K_D$ is ample iff $\deg(D) > n+1$. - If $P$ is smooth Fano, - $\Pic(X_P) = \ZZ^N$ where $N \da \size \mathrm{Vert}(P) - \dim (P)$. - $h^0(-K_X) = \size(M\intersect P)$. - $(-K_X)^n = n! \cdot \vol(P)$. --- - $h^{i, j} = \dim H^i(\Omega^j)$. - Constructing $\AA^n\times \GG_m^\ell$: take $\sigma\dual \da \Cone(e_1, \cdots, e_n) \union \Cone(\pm e_{n+1}, \cdots, \pm e_{n+\ell}) \subseteq \RR^{n + \ell}$. - $\CC[t^{\pm 1}] = \CC[t]_{\gens{t}}$. - Disc analogies: - Disc: $\CC[t]$ - Punctured disc: $\CC[t, t\inv]$. - Formal punctured disc: $\CC\fps{t}$. - Formal disc: $\CC\rff{t}$. - Example of computing the polar dual of $P_{-K_{\PP^2}}$: ![[2023-08-01 Old Notes 2023-08-25 16.12.11.excalidraw]] - If $P = \mathrm{Conv}\qty{{1\over a_F} u_F}$ then $P$ is reflexive $\iff a_F = -1$ for all $F$. - In dimension 2 - 16 reflexive polytopes - 5 of them smooth [^1]: What does this mean???