Root systems and periods on Hirzebruch surfaces (Q1311661)

Let \(\Sigma_ n = \mathbb{P} ({\mathcal O}_{\mathbb{P}^ 1} \oplus {\mathcal O}_{\mathbb{P}^ 1} (n))\) be the \(n\)-th Hirzebruch surface and let \(X\) be the surface obtained by blowing up \(n\) (distinct) points on \(\Sigma_ n\). In this paper the relation between surfaces like \(X\) and root systems of type \(A_{n-1}\) is investigated, in analogy with what happens with the cubic surfaces in \(\mathbb{P}^ 3\) and the root systems of type \(E_ 6\). The main results are the following: The structure of the root system in \(\text{Pic} X\) is used to prove a Torelli theorem for pairs \((X,K)\), where \(K\) is a certain anticanonical divisor. A family \(p : {\mathfrak X} \to S\) of surfaces of type \(X\) is constructed, where \(S\) is the quotient space of a maximal torus of the simple Lie group of type \(A_{n-1}\) by its Weyl group. As it happens for Del Pezzo surfaces and simple surface singularities of type \(E\), it is possible to view the fiber \({\mathfrak X}_ t\) of \(p\) as a compactification of the fiber of the semi-universal deformation of the simple surface singularity of type \(A_{n-1}\).