Let \(X\) be a smooth complete toric surface with \(d = \size \Sigma(1)\) rays and boundary divisors \(D_1, \ldots, D_d\) in cyclic order. Then \[ K_X^2 = 12 - d . \]
Write \(D_i^2 = -a_i\) with \(\sum_i a_i = 3d - 12\), and \(K_X = -\sum_i D_i\). Adjacent divisors meet once and non-adjacent ones not at all, so the cross terms contribute \(2d\): \[ K_X^2 = \Big( \sum_i D_i \Big)^2 = \sum_i D_i^2 + 2 \sum_i D_i \cdot D_{i+1} = -\sum_i a_i + 2d = -(3d - 12) + 2d = 12 - d . \]
This is Noether’s formula made trivial. A toric surface is rational, so \(\chi(\OO_X) = 1\), and its topological Euler characteristic is the number of maximal cones, which for a complete surface is also \(d\). Noether’s \(K^2 + \chi_{\mathrm{top}} = 12 \chi(\OO_X)\) then reads \(K^2 + d = 12\).
The checks are immediate. \(\PP^2\) has \(d = 3\) and \(K^2 = 9\). Each Hirzebruch surface has \(d = 4\) and \(K^2 = 8\), independent of \(a\). Every blowup adds one ray and drops \(K^2\) by one, so the toric del Pezzos \(\Bl_1, \Bl_2, \Bl_3 \PP^2\) have \(K^2 = 8, 7, 6\) and the list is exactly the surfaces with \(d \leq 6\) and \(-K\) ample.
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T-TORSURFSmooth complete toric surfaces and their intersection numbersuses