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• Tags:
  • #todo/untagged
• Refs:
  • http://jesusmartinezgarcia.net/wp-content/uploads/2019/08/toricNutshell.pdf#page=7
• Links:
  • toric variety
  • examples of varieties

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 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:
  • 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?
  • 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 http://jesusmartinezgarcia.net/wp-content/uploads/2019/08/toricNutshell.pdf#page=14
• 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 http://jesusmartinezgarcia.net/wp-content/uploads/2019/08/toricNutshell.pdf#page=25
• 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 \({\mathcal{O}}_{X, D}\)?
  • 
• 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:
• What is a \({\mathbb{Q}}{\hbox{-}}\)Cartier divisor?
  • 
• 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.
• 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:
• Give a formula for \(\operatorname{Div}(\chi^m)\).
  • Spoiler:
• 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 \({\mathbb{R}}^n\):

    

    Link to Diagram

• 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:

    

Link to Diagram - [ ] 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: - [ ] 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:
• What is a face of a lattice polytope?
  • Spoiler:
• 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:

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.

• 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:
  • 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:

  • 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:
  • 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

• 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 \(\Sigma=\left\{\sigma=\operatorname{Cone}\left(e_{2}, 2 e_{1}-e_{2}\right), \text { Cone }\left(e_{2}\right), \text { Cone }\left(2 e_{1}-e_{2}\right), 0\right\}\)

  • Spoiler:

  

Link to Diagram

• Work out the following example for \(X_\Sigma = {\mathbb{P}}^1\times {\mathbb{P}}^1\):

• 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?