Exercises In Log Geometry From Sam Johnston

This one actually confused me for a while, and it was because I didn’t read the question carefully: this is definitely not saying that the tropicalization \(\Sigma(X)\) recovers the fan of \(X\); it only recovers the cone complex associated to the fan. The cone complex throws away a decent bit of information; for instance, the cones \(\cone(e_1,e_2)\) and \(\cone(e_1 + de_2, e_1)\) in \(N = \mathbb Z^2\) are isomorphic as cone complexes but produce non-isomorphic toric varieties.

Let’s start with the affine case. Let \(X\) be the affine toric variety associated to \(\sigma\subset N_\mathbb R\) some strictly convex rational polyhedral cone in a lattice \(N_\mathbb R\), generated as a cone by primitive rays \(\rho_1,...,\rho_d \in N\). Then

• set \(m_i\) to be an element of \(\sigma^\vee\) such that \(\langle m_i, \rho_i\rangle = 0\) and \(\langle m_i, \rho_j\rangle \geq 0\) for all other \(j\)
• the toric divisor is then \(D = D_1 + ... + D_d\) where \(D_i = V(\chi^{m_i})\)
• the sheaf of monoids is \(\mathcal M = j_*\mathcal O^\times_{X\setminus D} \cap \mathcal O_X = \mathbb C^\times \cup \{\text{monomials in }\chi^{m_i}\}\) by the definition of the divisorial log structure
• the ghost sheaf is \(\overline {\mathcal M} = \{\text{monomials in }\chi^{m_i} \text{ up to scaling}\}\).

Let \(\eta_{\beta}\) be the generic point of the intersection \(D_{\beta_1}\cap ... \cap D_{\beta_j}\) for a \(j\)-subset \(\beta \in \binom{[d]}{j}\) of \([d]\). (Note that many of these intersections may be smaller than expected, take the cone over a pentagon for instance. Two divisors corresponding to rays which aren’t part of a common face will intersect to give a point rather than a line.) Then \[\overline{\mathcal M}_{\eta_\beta} = \{\text{monomials in }\chi^{m_{\beta_1}},...,\chi^{m_{\beta_j}} \text{ up to scaling}\}\cong \mathbb N^\beta \cong \mathbb N^j.\] If \(\eta_\beta \in \overline{\{\eta_\gamma\}}\), i.e. if \(\gamma\) is a “bigger” generic point, then \(\gamma \subset \beta\) and the generization map \(\overline{\mathcal M}_{\eta_\beta} \to \overline{\mathcal M}_{\eta_\gamma}\)is projection throwing away those coordinates of \(\beta\) not appearing in \(\gamma\). Dualizing gives us face maps \[\sigma_{\gamma} \to \sigma_{{\beta}}\] where \(\sigma_{\beta} := \Hom(\overline{\mathcal M}_{\eta_\beta}, \mathbb R_{\geq 0})\). Thus, the tropicalization process gives us

• one cone \(\sigma_\beta\)for each non-empty intersection \(D_{\beta_1}\cap ... \cap D_{\beta_j}\)
• a face map \(\sigma_\gamma \to \sigma_\beta\) whenever \(\eta_\beta\) is a specialization of \(\eta_\gamma\) (and hence is a larger cone in the fan)

This exactly recovers the cone complex structure of \(\sigma\), our original cone.

In the general case of a toric variety \(X\) associated to a fan, let \(\sigma_1,...,\sigma_n\) be the collection of maximal cones. Tropicalization produces the same system of cones and maps as described above as well as additional face maps identifying the maximal cones along common faces.

Note first that since \(X\) is a fine log scheme for each \(\sigma_\eta \in \Sigma(X)\) we can find an embedding \[\sigma_\eta = \Hom_{mon}(\overline{\mathcal M}_\eta, \mathbb R_{\geq 0}) \hookrightarrow \Hom_{grp}(\overline{\mathcal M}_\eta^{grp}, \mathbb R) = \mathbb R^n.\] This embedding exists because fine log structures are integral and the \(n\) is finite since fine log structures are locally finitely generated. We hence identify \(\sigma_\eta\) with its embedding inside the real vector space \(\mathbb R^n\), hence it makes sense to talk about a “linear function on \(\sigma\)”. Note also that every linear function on \(\sigma\) is the restriction of a linear function \(\mathbb R^n \to \mathbb R\) to \(\sigma\).

By a “piecewise linear function on \(\Sigma(X)\)” we mean a continuous function \(f:|\Sigma(X)|\to \mathbb R\) so that

• the restriction \(f|_\sigma\) is linear for each \(\sigma \in \Sigma(X)\) and
• the linear functions comprising \(f\) agree along faces – this is compatibility with gluing.

Let’s start with a single cone \(\sigma = \Hom(\overline{\mathcal M}_\eta, \mathbb R_{\geq 0})\). By our initial remarks, a linear function on \(\sigma\) is an element \(f\in \Hom(\Hom(\overline{\mathcal M}_\eta^{grp}, \mathbb R), \mathbb R).\) We have isomorphisms \[\Hom(\overline{\mathcal M}_\eta^{grp}, \mathbb R)\cong \Hom(\overline{\mathcal M}_\eta^{grp}, \mathbb Z)\otimes_{\mathbb Z}\mathbb R = (\overline{\mathcal M}^{grp}_\eta)^\vee \otimes_{\mathbb Z} \mathbb R\] so \[\begin{aligned} \Hom(\Hom(\overline{\mathcal M}_\eta^{grp}, \mathbb R), \mathbb R) &\cong \Hom((\overline{\mathcal M}^{grp}_\eta)^\vee \otimes_{\mathbb Z} \mathbb R, \mathbb R) \\ &\cong (\overline{\mathcal M}^{grp}_\eta)^{\vee\vee} \otimes_{\mathbb Z} \mathbb R \\ &\cong \overline{\mathcal M}^{grp}_\eta \otimes_{\mathbb Z}\mathbb R.\end{aligned}\]

In the case that \(\Sigma(X) = \sigma_\eta\), the stalk \(\overline{\mathcal M}_\eta\) is the global sections, and hence we get that linear functions on \(\sigma\) are precisely \(\overline{\mathcal M}^{grp}_\eta \otimes_{\mathbb Z}\mathbb R\).

In the general case, a piecewise linear function \(f\) on \(\Sigma(X)\) is, once again, a collection of linear functions on the cones comprising \(\Sigma(X)\) which agree on common faces. Using the identification we established above, this is precisely the data of a global section of \(\overline{\mathcal M}_X^{gp}\) base changed to \(\mathbb R.\)

The relevant portion of the long exact sequence in cohomology gives us \[H^0(S, \mathcal M^{gp}_S) \to H^0(S,\overline{\mathcal M}^{gp}_S) \xrightarrow{f} H^1(S, \mathcal O^\times_S).\] From Exercise 2 we get that \(H^0(S,\overline{\mathcal M}^{gp}_S)\) is the space of integral real-valued piecewise linear functions on \(\Sigma(S)\). Since \(H^1(S,\mathcal O^\times_S)\cong \text{Pic}(S)\), \(f(\rho)\) can be interpreted as a line bundle on \(S\), which we call \(\mathcal O(\rho)\). When \(\rho\) is non-negative on all of \(\Sigma(X)\), it is a global section of \(H^0(S, \overline{\mathcal M}_X)\) (without the groupification).

Everything in sight is affine, with toric log structure. The map on rings is \[f^\sharp:f^{-1}\mathcal O_S\to \mathcal O_C, t\mapsto t\] and since the sheaf of monoids of a divisorial log structure sits inside the structure sheaf, \(f^\sharp\) deteremines \(f^\flat\):

On global sections, \(\mathcal M_S(S) = \mathbb C^\times \cup \{\text{monomials in }t\}\) and \(\mathcal M_C = \mathbb C^\times\cup \{\text{monomials in }\overline x, \overline y \text{ and } \overline t\} \), where \(\overline x\) denotes the image of \(x\) under \(\mathbb C[x,y,t] \to \mathbb C[x,y,t]/(xy - t^k)\) for example, and hence the map is \(f^\flat:t\mapsto t.\)

A comment about \(\mathcal M_C\): the toric divisor in \(C\) is \(V(\overline x, \overline t)+V(\overline y, \overline t)\), viewing \(x\), \(y\) and \(t\) as monomials in \(\mathcal O_C\). This can be seen by viewing \(C\) inside \(\mathbb A^3_{xyt}\) with torus \(T = \{(t_1,t_1^{-1}t_2^{k}, t_2)\}\) for \(t_1,t_2 \neq 0\) and then checking that these two terms are indeed that the toric invariant divisors. If a prime ideal in \(\mathcal O_C\) contains \(\overline x\), then it contains \(\overline {x}\overline{y} = \overline t^k\) and hence also contains \(\overline t\), and symmetrically for \(\overline y\), so we can actually write this divisor as \(V(\overline x) + V(\overline y)\). Likewise, if a prime ideal contains \(\overline t\), then it contains \(\overline t^k = \overline x\cdot \overline y\) and thus contains both \(\overline x\) and \(\overline y\). Geometrically, we can see this because the intersection of \(C\) and the hyperplane \(V(x)\) in \(\mathbb A^3_{xyt}\) is precisely \(V(\overline x)\).

The intersection of \(V(x)\cup V(y)\) with \(C\) in \(\mathbb A^3_{xyt}\) is \(V(t^k)\cap C\), a fat plane intersect \(A^3_{xyt}\).

Ghost sheaves on global sections:

• \(\overline {\mathcal M_C} =\text{monomials in }\{\overline x, \overline y, \overline t\} \text{ w/ relation that } \overline x + \overline y = \overline kt.\)
• \(\overline{\mathcal M}_S = \text{monomials in }\overline t\)

Homs:

• \(\sigma_C = \Hom(\overline{\mathcal M}_C, \mathbb R_{\geq 0})\), each map determined by image of \(\overline x\) and \(\overline y\) in \(\mathbb R_{\geq 0}\). Given a map \(\varphi\) we get that \(\varphi(\overline t) = \varphi(\overline x + \overline y)/k\). Hence \(\sigma_C\cong \mathbb R_{\geq 0}^2\)
• \(\sigma_S = \Hom(\overline{\mathcal M_S}, \mathbb R_{\geq 0})\), each map determined by image of \(\overline t\). Hence \(\sigma_S \cong \mathbb R_{\geq 0}\).

The map \(\Sigma(f): \sigma_C \to \sigma_S\) is therefore \(\varphi\mapsto (\overline t \mapsto \frac{\varphi(\overline x + \overline y)}{k})\). The fiber of \(1\) over this map is, for instance, the line connecting \((k,0)\) to \((0,k)\) in \(\sigma_S\).

1. A formally smooth morphism is one in which every diagram as above with \(\overline T \to T\) a square-zero extension of affines admits a dashed line making the diagram commute. Consider the case where
2. \(X = \Spec \mathbb C[x,y,t]/(xy-t^k)/\) for some \(k \geq 2\),
3. \(Y = \Spec\mathbb C[t]\) and
4. \(f\) is the map corresponding to the ring map \(\mathbb C[t]\xrightarrow{t\mapsto t} \mathbb C[x,y,t]/(xy - t^k)\).

The intuition is that \(f\) is not smooth since at \(t = 0\), the fiber of \(f\) is the union of coordinate lines in \(\mathbb A^2_{\mathbb C}\). Set

• \(\overline T = \Spec \mathbb C[\epsilon]/\epsilon^2\)
• \(T = \Spec \mathbb C[\epsilon]/\epsilon^3\)
• \(\overline T\to T\) to be the morphism corresponding to the truncation map \(\mathbb C[\epsilon]/\epsilon^3 \to \mathbb C[\epsilon]/\epsilon^2\).

Now consider the tangent vector of \(X\) corresponding to the ring morphism \(\mathbb C[x,y,t]/(xy-t^k) \to \mathbb C[\epsilon]/\epsilon^2\) given by \(x\mapsto \epsilon, y\mapsto \epsilon, t\mapsto 0\). This preserves the relation \(xy = t^k\) since \(\epsilon^2 = 0\) in \(\mathbb C[\epsilon]/\epsilon^2\). We get the following two commutative squares, and since everything in sight is affine, the rightmost square determines the scheme theoretic leftmost square:

The desired lift \(g\) must satisfy

• \(g(x) = \epsilon + a\epsilon^2\),
• \(g(y) = \epsilon + b\epsilon^2\) and
• \(g(t) = 0\)

for some \(a,b\in \mathbb C\) in order to preserve commutativity. But then \[g(xy) = \epsilon^2 + a\epsilon^3 + b\epsilon^3 + ab\epsilon^4 = \epsilon^2 \neq 0\] in \(\mathbb C[\epsilon]/\epsilon^3\), so \(g\) cannot be a ring morphism. Thus there exists no lift \(g\), and hence the morphism \(f\) is not formally smooth.

1. A log morphism \(f:X\to Y\) is log smooth if it is induced etale locally by a morphism of monoids + a technical condition; i.e. if we have a diagram

where \(g\) is induced by morphism \(Q\to P\) and the induced map \[X\to \Spec \mathbb Z[P]\times_{\Spec \mathbb Z[Q]} Y\] is formally smooth.

In our case, \(X = \Spec \mathbb C[P]\) for the monoid \(P = \cone(e_1, e_2, ke_1 - e_2)\cap \mathbb Z^2\subset \mathbb R^2\) and \(Y = \Spec\mathbb C[\mathbb N]\), so we obtain the square automatically. The relevant induced map is \[X\to \Spec \mathbb Z[P] \times_{\Spec \mathbb Z[\mathbb N]}Y.\] Given any morphism \(\varphi:S\to T\) of schemes, \(T\times_T S\cong S\) (at least when we take the map \(T\to T\) to be the identity). Since \(Y \cong \Spec \mathbb Z[\mathbb N]\times_{\Spec\mathbb Z}\Spec\mathbb C\), we have

Hence the induced map is just the identity \(X\to X\), which is always formally smooth. See also Example 3.24 in Mark Gross’s book, it’s a slight generalization of this problem.

Let \(f:T\to S\) be a log map. This map is determined scheme theoretically entirely by \[f^\sharp: \mathbb C = f^{-1}\mathcal O_S \to \mathcal O_T = \mathbb C,\] since there is only one option for the topological map. If we work in the category of schemes over \(\mathbb C\), then this ring map must also be \(\mathbb C\)-linear in which case the only option is that this is the identity.

The map of log structures \(f^\flat:f^{-1}\mathcal M_S\to \mathcal M_T\) is determined by the map on global sections, so we need only describe a map \[f^\flat:\mathbb C^\times \oplus Q \to \mathbb C^\times \oplus \mathbb N.\] This must fit into the following commutative diagram

encoding compatibility with \(f^\sharp\) and the log structure maps, hence \(f^\flat|_{\mathbb C^\times \oplus \{0\}}\) must be the restriction of \(f^\sharp\) to \(\mathbb C^\times\).

Therefore the only non-determined portion of the morphism \(f\) is \(f^\flat|_{\{1\}\oplus Q}:Q\to \mathbb C^\times \oplus \mathbb N\). The only restriction here is that no nonzero elements of \(Q\) can be mapped to \(\mathbb C^\times \oplus \mathbb \{0\}\}\), since we must have \(\alpha_s(f^\flat(0,q)) = 0\) by the commutativity of the above diagram.

We can say a little more: since \(Q\) is a toric monoid, it is the intersection of a cone \(\sigma\subset Q^{gp}\otimes_{\mathbb Z}\mathbb R\) with \(Q^{gp}\), and thus the map \(\overline f^\flat:Q\to \mathbb N\) is the restriction of a linear map \(Q^{gp}\to \mathbb Z\) which is non-negative on \(Q\).