Smooth complete toric surfaces and their intersection numbers

A smooth complete toric surface is encoded by its cyclic fan: adjacent divisors meet once, nonadjacent ones miss, and $D_i^2=-a_i$.Suggest a slogan

(Classification)

A smooth complete toric surface is given by rays $v_0, v_1, \ldots, v_{d-1}, v_d = v_0$ in counterclockwise order such that each consecutive pair $\ts{v_i, v_{i+1}}$ is a basis of $N \cong \ZZ^2$. Consecutive triples then satisfy a unique relation \[ v_{i-1} + v_{i+1} = a_i v_i, \qquad a_i \in \ZZ , \] and these integers are constrained by \[ \prod_{i} \begin{bmatrix} 0 & -1 \\ 1 & a_i \end{bmatrix} = \id , \qquad \sum_i a_i = 3d - 12 . \] For $d = 3$ the surface is $\PP^2$; for $d = 4$ it is a Hirzebruch surface $\FF_a$; for $d \geq 5$ it is a blowup of $\PP^2$ or of some $\FF_a$ at torus-fixed points.

(Intersection numbers)

With the same labelling, \[ D_i \cdot D_j = \begin{cases} 0 & \abs{i - j} > 1 , \\ 1 & \abs{i - j} = 1 , \\ -a_i & i = j . \end{cases} \]

Two adjacent rays span a cone, whose fixed point is the single transverse intersection of the two divisors, giving the $1$. Non-adjacent rays span no cone, so the divisors are disjoint, giving the $0$. The self-intersection comes from $\operatorname{div}(\chi^m) \cdot D_i = 0$ for every $m$: choose $m$ with $\inp{m}{v_i} = -1$ and $\inp{m}{v_{i\pm1}}$ determined by the relation, and $D_i^2 = -a_i$ drops out.

The sum rule $\sum a_i = 3d - 12$ is a useful check. For $\PP^2$, $d = 3$ and each $a_i = -1$, so each line has $D_i^2 = 1$. For $\FF_a$, $d = 4$ and the $a_i$ are $a, 0, -a, 0$, giving self-intersections $-a, 0, a, 0$.

The self-intersections $D_i^2 = -a_i$ are the whole intersection form of the surface, since the $D_i$ generate $\Pic$ and $D_i \cdot D_j$ is $1$ for adjacent rays and $0$ otherwise. They decide the birational geometry: a boundary curve with $D_i^2 = -1$ is a $(-1)$-curve, contracting it removes the ray $v_i = v_{i-1} + v_{i+1}$, and repeating reaches $\PP^2$ or a Hirzebruch surface $\FF_a$, which is the classification of smooth complete toric surfaces. Nefness and ampleness are also read off: a torus-invariant divisor on a complete toric surface is nef exactly when its intersection with every $D_i$ is nonnegative.

Classify the smooth complete toric surfaces.

How do you compute intersection numbers of the boundary divisors of a toric surface?

Why should someone care about the self-intersection of invariant divisors?

Dependencies

T-TORDIVDivisors, class group and Picard group from the raysuses

Backlinks

FE-TORBLOWBlowup as star subdivision, and P^2 blown up at a fixed pointuses
PR-FULKSQK squared equals twelve minus the number of rays on a smooth complete toric surfaceuses
FE-TORHIRZThe Hirzebruch surface F_a from its fanuses

What links to this

Toric surfaces and toric morphisms