Given a complete smooth Toric surface (over $\mathbb C$), is its intersection form well-known? Or is there an algorithm to calculate it? Thanks in advance.
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2$\begingroup$ Have a look at Fulton's Introduction to toric varieties, §5.2. $\endgroup$– R. van Dobben de BruynCommented Aug 15, 2022 at 22:11
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1$\begingroup$ I would like to point out that from an algebraic view point the intersection forms that appear are extremely restricted. There is the intersection form of $S^2 \times S^2$ $(a,b).(c,d) = ad+bc$ which is the only even form which appears. All the rest are direct sum of one copy of the standard positive Euclidean form and some number of copies of the negative standard Euclidean form. (that is the intersection form of the blow up of $\mathbb{CP}^2$ in $k$ points). So, if the toric surface has $b_2>2$ then the intersection form is determined up to isomorphism by $b_2$. $\endgroup$– Nick LCommented Aug 20, 2022 at 13:29
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The Picard group of a toric surface is generated by the simple toric divisors $D_1,D_2,\dots,D_n$, and if $v_1,v_2,\dots,v_n$ are the generators of the rays of the corresponding fan, there are relations $$ \sum f(v_i)D_i = 0 $$ for all linear functions $f$. The intersection product is determined by these linear relations and simple intersection formulas $$ D_i \cdot D_j = \begin{cases} 1, & \text{if $|i-j| = 1$},\\ 0, & \text{if $|i-j| > 1$}, \end{cases} $$ where the indicies are assumed to by ordered cyclically (and to compute $D_i \cdot D_i$ one should use the linear relations).
