Toric Surfaces from Fans

Here are examples of 2-dimensional toric varieties with all the affine spectra worked out and the gluing data specified explicitly to the degree necessary to work out e.g. Čech cohomology by hand. I’ve used sage at times, and will include the code I’ve utilized. I’ll also include some visualizations of the cones when possible. I use $\hull(\{v_i\})$ to denote the convex hull of the $v_i$, and $\cone(\{v_i\})$ to denote the set $\{a_1v_1+a_2v_2+\cdots\ \text{s.t.}\ a_i\in \mb R_{\geq 0}\}$. If $\sigma$ is a cone, $\sigma^\nu = \{v\in\mathbb R^n \text{ s.t. } v\cdot w\geq 0\ \ \forall w\in \sigma\}$ is the dual of $\sigma$.

A fan from the vectors $\left\{\mat{1\\2}, \mat{-2\\-1}, \mat{3\\-1}\right\}$

Define a fan $F = \fan(\sigma_1, \sigma_2, \sigma_3)$ for cones $\sigma_i$ as defined below. Compute their duals $\sigma_i^\nu$ and generators for the intersection of their duals with $\mb Z^2$. All of this can be done more or less by inspection.

Compute the toric ideals associated with each cone. The naive way to do this is to find all $\mb Z$-linear combinations of the $\mb N$-generators which evaluate to zero (this can be accomplished by solving an appropriate liner equation, or row reducing an appropriate matrix). We then write such vectors as $l = l^+-l^-$ and consider the ideal generated by expressions of the form $x^{l^+}-x^{l^-}$ (using multi-indexing). It is easy to find generators for such an ideal for $\sigma_2$ and $\sigma_3$. First we compute the set of all linear combinations which evaluate to zero, which we call $Z$. This is the kernel of the natural map taking $\mb N^n\to S = \sigma^\nu_i\cap \mb Z^2$. There’s only one generator in each case, which means that every vector $(a_1, ..., a_i)$ such that $a_1g_1+...+a_ig_i=0$ is an integer multiple of that generator. But note that $(a-b)^2 - 2b(a-b) = a^2-2ab+b^2 -2ab-2b^2 = a^2-b^2$, and $x^{(-l)^+}-x^{(-l)^-} = -(x^{l^{+}}-x^{l^{-}})$, so the entire ideal is generated by $x^{l^+}-x^{l^-}$. Clearly this ideal is the kernel of the natural induced map $k[\mb N^n]\to k[S]$ coming from the map discussed above, and so we can use it to express $k[S]$ as the quotient of a polynomial ring.

This method doesn’t work as well for $\sigma_1$; it turns out the set $Z$ has three generators, and it’s difficult to pick out generators for the corresponding ideal by hand. Instead, we examine the embedding of the torus $\mb T^2$ into $\mb A^2$ induced by the map $k[x, y, z, w, v]\to k[t^{\pm 1}, u^{\pm 1}]$ given by the rules to the left.

• $x\mapsto t^2u^{-1}$
• $y\mapsto tu^3$
• $z\mapsto t$
• $w\mapsto tu$
• $v\mapsto tu^2$

The graph of this map is the variety given by the ideal

we wish to find the smallest algebraic variety containing the projection of this graph to $\spec k[x, y, z, w, v]$. This is given by elimination theory by computing a Groebener basis for $I$ with respect to an elimination ordering for $k[x, y, z, w, v, t, u]$, and then selecting only those terms which do not contain $t$ and $u$. The elimination ordering I used was the lexicographical ordering with $t$ and $u$ ordered before the rest. I computed this Groebener basis with sage; here’s the code.

R.<t, u, x, y, z, w, v> = PolynomialRing(CC, 7, order="lex") 
Rprime.< x, y, z, w, v> = PolynomialRing(CC, 5, order="lex") 
S = Localization(R, ('t', 'u'))
I = ideal(u*x - t^2, y - t*u^3, z - t, w - t*u, v - t*u^2)
B = I.groebner_basis(); print("The Groebner Basis is \\[", latex(B), "\\]")
C = B[5:]; print("Our generators for the elimination ideal are: \\[", latex(C), '\\]')
J = ideal(Rprime, C)
print("Is $", latex(J), "$ prime: ",  J.is_prime())

This gives us that the Groebner Basis is

and our generators for the elimination ideal are

Is $\left(x y - z w^{2}, x w - z^{3}, x v - z^{2} w, y z - w v, y w - v^{2}, z v - w^{2}\right)$ prime: True

This gives us our final ideal, and so we have $k[S]$ isomorphic to

This gives the three open affines for our scheme. We must now specify the gluing data. We will glue along three open subsets $V_1$, $V_2$, and $V_3$; we will denote the affine scheme associated to $\sigma_i$ by $U_i$. We then have (when there’s an ambiguity as to which side a cone is on, take the cone going clockwise from the first vector to the second).

Writing the $\mb N$-generators for each of the $U_i$ in terms of the $\mb N$-generators for each of the $V_i$ yields maps as defined in the table below. The universal property of localization then gives factorizations of these maps through certain localizations of the source ring, which are isomorphisms.

I leave the well definedness of these maps and the fact that they are open immersions to the reader. Hint: They should be well defined (loosely) because they “come from” the same relations on the fan as the quotients. The open immersion follows from the fact that they induce isomorphisms from a localization of the source ring.

We now know that our scheme $X$ is glued together from the schemes

Perhaps intersting to note is that for $k=\mb R$, $U_2$ and $U_3$ look like the following (about the origin).

These look about the same, as is to be expected. Moreover, one can easily see here that they are birational to $\spec k[a^{\pm 1}, b^{\pm 1}]$, or $\mb A - \ell - \ell'$ for lines $\ell, \ell'$ passing through the origin. They also admit a fibration over $\mb A^1$ who’s fibers are the degeneration of a hyperbola to a pair of lines. Taking the intersection of $U_2$ with a hyperplane of the form $V(az-by)$ yields (so long as both $a$ and $b$ are nonzero) $k[x, z] / (\frac baz^2-x^3)$, a transformation of the cuspodial cubic; a similar cross-section of $U_3$ yields the cuspodial quintic. This gives a fibration $U_i\to \mb P_1$ who’s generic fibers are the cuspodial cubic or quintic and who’s fibers at two points are lines with some multiplicity. Finally, it admits one last fibration over $\mb A^1$ (technically two, I suppose, but they’re identical): cutting with the hyperplane given by $V(y-a)$ or $V(z-a)$. This yields the degenration of a cubic embedding of the line in a surface to a degree-one embedding of a line with multiplicity.

Finally, we note that the intersections of the gluing patches are all isomorphic to $\spec k[x^{\pm 1}, y^{\pm 1}] \cong \mb T^2$. This is enough data to compute Čech cohomology, and we have acheived our goal.