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• Refs:
  • Brasselet: https://www2.math.ethz.ch/education/bachelor/lectures/fs2015/math/alg_geom/brasselet
  • Danilov: https://www.maths.ed.ac.uk/~v1ranick/papers/danilov.pdf
  • Popma: https://www.math.ru.nl/~bosma/Students/GeertPopmaMSc.pdf
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exercises in toric geometry

Source: http://ibykus.sdf.org/website/lang/en/toric-old.php?lang=en

Zariski toplogy 1. What are the open sets in \(\mathbb{A}^{1}\) ? 2. Show that the Zariski toplogy on \(\mathbb{A}^{1} \times \mathbb{A}^{1}\) is different from the product topology of the Zariski topologies of the factors 3. What is the Zariski closure of \(\left\{x_{2}=e^{x_{1}}\right\} \subset \mathbb{A}^{2}\) ? Intrinsic definition of Spec More generally one defines the spectrum \(X=\operatorname{Spec} R\) of an integral domain as the set of its prime ideals. With closed subsets of the form \(V(I):=\{\mathfrak{p} \in \operatorname{Spec} R \mathrel{\Big|}I \subset \mathfrak{p}\}\). The sheaf of regular function is given by \(\mathcal{O}(U)=\{f / g \in Q(R) \mathrel{\Big|}\forall p \in U: g \notin \mathfrak{p}\}\) (there is even a version for general rings). 1. Show that this defines actually a topology 2. Show that there is a homeomorphism between the subset of maximal ideals \(\operatorname{Specm}(P)=\operatorname{Spec}(P)\) with \(P=\mathbb{C}\left[x_{1}, \ldots, x_{m}\right]\) (subset topology) and \(\mathbb{A}^{m}\). 3. Observe that there is a natural restriction of \(\mathcal{O}\) to the subset of maximal ideals and a canonical algebra isomorphism between \(\mathcal{O}_{A^{\mathrm{m}}}(U)\) and \(\mathcal{O}_{\text {Specm }(P)}(U)\) 4. Use this to show that the same holds for Specm \(\left[x_{1}, \ldots, x_{m}\right] / p\) and \(V\) (p) for every prime ideal \(\mathfrak{p}\) Semigroups and semigroup algebras Give \(k[S]\) in terms of generators and relations and calculate the singular locus of the corresponding variety for the following semigroups: 1. \(S=\mathbb{Z}_{>0} \backslash\{1\}\) 3. \(S=\mathbb{Z}_{\geq 0}^{2} \backslash\{(1,0)\}\). Solution: 1. \(k\left[x_{1}, x_{2}\right] /\left(x_{1}^{3}-x_{2}^{2}\right)\) 2. \(k\left[x_{1}, x_{2}, x_{3}\right] /\left(x_{1} x_{2}^{2}-x_{3}^{2}\right)\) 3. \(k\left[x_{1}, x_{2}, x_{3}, x_{4}\right] /\left(x_{1} x_{3}-x_{2} x_{4}, x_{1}^{3}-x_{2}^{2}, x_{4} x_{2}-x_{3} x_{1}^{2}, x_{4}^{2}-x_{1} x_{3}^{2}\right)\) Binomial algebras and semigroup algebras 1. Find the semi-group \(S\) for the binomial algebra \(k[x, y, z, w] /\left(x z-y^{2}, y w-z^{2}\right)\). Hint: start with constructing the map \(\mathbb{Z}^{4} \rightarrow M\) 2. Proof that every algebra \(k\left[t_{1}, \ldots, t_{\ell}\right] /\left(t^{a_{1}}-t^{b_{1}}, \ldots, t^{a_{m}}-t^{b_{m}}\right)\) without zero-divisors arises as a semigroup algebra. 3. Constructing algebras with zero divisors: consider the semigroups \(S_{1}, S_{2}\) in \(M=\mathbb{Z}^{2}\) being generated by \(\{(0,1)\), (1, 1) \(\}\) and \(\{(1,0),(1,1)\}\), respectively. Glue \(k\left[S_{1}\right]\) and \(k\left[S_{2}\right]\) along \(k\left[S_{1} \cap S_{2}\right]\) as follows. Consider the algebra \(A=\bigoplus_{u \in S_{1} \cup S_{2}} k \cdot \chi^{u}\) with multiplication defined by \begin{align*} \chi^{u} \cdot \chi^{w}= \begin{cases}\chi^{u+w} & , u, w \in S_{1} \text { or } u, w \in S_{2} \\ 0 & , \text { else }\end{cases} \end{align*} Describe \(\operatorname{Spec}\left(k\left[S_{1}\right]\right), \operatorname{Spec}\left(k\left[S_{2}\right]\right), \operatorname{Spec}\left(k\left[S_{1} \cap S_{2}\right]\right)\) and \(\operatorname{Spec}(A) .\)

Normality 1. Show that the normalisation is given by the saturation of the semigroup \(S^{\text {sat }}=\{u \mathrel{\Big|}m u \in S, m \in \mathbb{N}\}\). 2. Obtain the normalisation map for the cusp in coordinates 3. Show that \(\operatorname{Spec}(k[S])\) for \(S=\mathbb{Z}_{>0}^{2} \backslash\{(1,0)\}\) is not normal, but non-singular outside a point 4. Show that \(\operatorname{Spec}(k[S])\) for \(S=\mathbb{Z}_{\geq 0}^{2} \backslash\{(1+2 \mathbb{Z}) \times 0\}\) is not normal and singular along a line (use Jacobian criterion) 5. How does the normalisation maps look like in coordinates? What happens to the singular line? The functor TV 1. Show that \(T V\left(\sigma \times \sigma^{\prime}, N \times N^{\prime}\right)=T V(\sigma, N) \times T V\left(\sigma^{\prime}, N^{\prime}\right)\). What does this imply for the case that \(\sigma^{\prime}=\{0\}\) ? 2. Calculate \(T V(\sigma, N)\) for \(\sigma=\operatorname{pos}\left(\begin{array}{ll}1 & 1 \\ 0 & n\end{array}\right)\) (here I actually mean the columns of the matrix) 3. Describe \(C_{Q}=T V(\sigma, N)\) for \(\sigma=\operatorname{pos}\left(\begin{array}{llll}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1\end{array}\right)\) 4. Consider the morphism induced by the inclusion of cones pos \(\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right) \subset \operatorname{pos}\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\). Show that this morphism is an isomophism onto its image outside a line, but this line gets contracted to a point. One can see this in coordinates as well as by the inclusion of semigroups (Remember the description of invariant closed subvarieties). 5. The example from the lecture: Let \(\sigma\) be the positive orthant in \(\mathbb{Q}\) and we consider the lattice inclusion \(N=\mathbb{Z} \subset \mathbb{Z}^{2}+\mathbb{Z} \cdot \frac{1}{2}(1,1)=N^{\prime}\). Now, the using the standard scalar product to identify \(N_{\mathbb{Q}}=N_{\mathbb{Q}}^{\prime}\) with its dual vector space we see the dual lattices by taking all vectors which give integers when pairing with lattice elements, this gives identifications \(M=N^{*}=\mathbb{Z}^{2}\) and \(M^{\prime}=\left(N^{\prime}\right)^{*}=2 \mathbb{Z}^{2}+\mathbb{Z} \cdot(1,1)\). Now, the inclusion \(\sigma^{\vee} \cap M^{\prime} \subset \sigma^{\vee} \cap M\) leads to an inclusion of algebras. Now, \(k\left[\sigma^{\vee} \cap M^{\prime}\right] \subset k\left[\sigma^{\vee} \cap M\right]\) can be seen as the ring of invariants under the \(\mathbb{Z} / 2 \mathbb{Z}\)-action given by \(\chi^{e_{i}} \mapsto-\chi^{e_{i}}\). Observe that this generalises to maps induced by lattice inclusions of finite index.

The functor TV 1. Show that \(T V\left(\sigma \times \sigma^{\prime}, N \times N^{\prime}\right)=T V(\sigma, N) \times T V\left(\sigma^{\prime}, N^{\prime}\right)\). What does this imply for the case that \(\sigma^{\prime}=\{0\}\) ? 2. Calculate \(T V(\sigma, N)\) for \(\sigma=\operatorname{pos}\left(\begin{array}{ll}1 & 1 \\ 0 & n\end{array}\right)\) (here I actually mean the columns of the matrix) 3. Describe \(C_{Q}=T V(\sigma, N)\) for \(\sigma=\operatorname{pos}\left(\begin{array}{llll}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1\end{array}\right)\) 4. Consider the morphism induced by the inclusion of cones pos \(\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right) \subset \operatorname{pos}\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\). Show that this morphism is an isomophism onto its image outside a line, but this line gets contracted to a point. One can see this in coordinates as well as by the inclusion of semigroups (Remember the description of invariant closed subvarieties). 5. The example from the lecture: Let \(\sigma\) be the positive orthant in \(\mathbb{Q}\) and we consider the lattice inclusion \(N=\mathbb{Z} \subset \mathbb{Z}^{2}+\mathbb{Z} \cdot \frac{1}{2}(1,1)=N^{\prime}\). Now, the using the standard scalar product to identify \(N_{\mathbb{Q}}=N_{\mathbb{Q}}^{\prime}\) with its dual vector space we see the dual lattices by taking all vectors which give integers when pairing with lattice elements, this gives identifications \(M=N^{*}=\mathbb{Z}^{2}\) and \(M^{\prime}=\left(N^{\prime}\right)^{*}=2 \mathbb{Z}^{2}+\mathbb{Z} \cdot(1,1)\). Now, the inclusion \(\sigma^{\vee} \cap M^{\prime} \subset \sigma^{\vee} \cap M\) leads to an inclusion of algebras. Now, \(k\left[\sigma^{\vee} \cap M^{\prime}\right] \subset k\left[\sigma^{\vee} \cap M\right]\) can be seen as the ring of invariants under the \(\mathbb{Z} / 2 \mathbb{Z}\)-action given by \(\chi^{e_{i}} \mapsto-\chi^{e_{i}}\). Observe that this generalises to maps induced by lattice inclusions of finite index. Toric morphisms and first examples of gluing 1. Given the cone \(\sigma=\operatorname{pos}\left(\begin{array}{llll}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1\end{array}\right)\) of \(C_{Q}\) as before. There is a toric map to \(\mathbb{A}^{1}\) being induced by the lattice homomorphism \(F\) given by projection to the third coordinate. What are then fibres of this morphism? What are the fibres of \(F_{\mathbb{Q}}\) seen are a mapping of cones (i.e. what does \(F_{\mathbb{Q}}^{-1}(v) \cap \sigma\) look like)? How could the geometry of both fibres be related to each other? 2. Consider the toric variety given by gluing the toric varieties corresponding to the rays \(\operatorname{pos}((1,0))\) and \(\operatorname{pos}((0,1))\) along \(T V(\{0\})\) and the toric morphism to \(\mathbb{P}^{1}\) given by the lattice homomorphism corresponding to the matrix \((1,-1)\). Describe the variety and the morphism in coordinates 3. Consider the two cones \(\sigma_{1}=\operatorname{pos}\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right)\) and \(\sigma_{2}=\operatorname{pos}\left(\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)\). Study the variety \(X\) obtained by gluing TV \(\left(\sigma_{1}\right)\) and TV \(\left(\sigma_{2}\right)\) along \(\mathrm{TV}\left(\sigma_{1} \cap \sigma_{2}\right)\) and the toric morphism \(X \rightarrow C_{Q}\) given by the identity (see above for the definition of the quadric cone \(C_{Q}\) ) Fans and toric morphisms 1. Consider the ideal \(\mathfrak{p}=I_{\rho} \subset k\left[\sigma^{\vee} \cap M\right]\) corresponding to a ray \(\rho \prec \sigma\). Show that \(k\left[\sigma^{\vee} \cap M\right]_{\mathfrak{p}}\) is a discrete valuation ring by stating the corresponding valuation of the function field of \(T V(\sigma)\). 2. Consider the fan \(\left\{\mathbb{Q}_{\geq 0}(1,0), \mathbb{Q} \geq 0(1,0), 0\right\}\) and the map of fans to the fan of \(\mathbb{P}^{1}\) given by \(\left(v_{1}, v_{2}\right) \mapsto v_{1}+v_{2}\). Describe the corresponding toric morphism. 3. Describe the embedding of \(Y=V\left(x_{1} x_{3}-x_{2}^{2}\right) \subset \mathbb{A}^{3}\) by a morphisms of cones (or fans if you wish) 4. Describe the blow up of \(\mathbb{A}^{3}\) in a point by a morphism of fans (try to make a guess by looking at the example of \(\mathbb{A}^{2}\) from the lecture). 5. Describe the normalisation (actually it is already normal) of the strict transform of \(Y\) under this blow by using the equivalence of the categories of normal toric varieties and fans and the universal property of the strict transform. To do this you can use the fact that the strict transform is toric again (but you can also derive this by using the embedding of tori coming with the embedding of \(Y\) ). 6. Show that the constructed morphism of fans induces a closed embedding. It follows, that the strict transform is indeed normal and given by the fan constructed above. 7. Realise, that we have constructed an embedded resolution of singularities.

The normal fan 1. What is the normal fan of the simplex \(\operatorname{conv}\left(0, e_{1}, \ldots, e_{n}\right) \subset M_{\mathbb{Q}}\) 2. What is the normal fan of \(\operatorname{conv}\left(e_{1}, \ldots, e_{n}\right)+\operatorname{pos}\left(e_{1}, \ldots, e_{n}\right)\) 3. Given a polyhedron \(P+\sigma\). Consider the function \(h_{\Delta}: \sigma^{\vee} \rightarrow \mathbb{Q}\) given by \begin{align*} h_{\Delta}(v)=\min \langle\Delta, v\rangle:=\min \{\langle u, v\rangle \mathrel{\Big|}u \in \end{align*} Show that this function is linear on every cone of the normal fan of \(\Delta\). The divisor class group 1. Calculate the class group of our standard example \(\left(\mathbb{P}^{1}\right)^{2} / \mu_{2}\) using toric methods 2. Show: every smooth complete toric variety has a free divisor class group