On minimal rational elliptic surfaces

Abstract

We construct $13$ projective $\mathbb{Q}$-factorial Fano toric varieties and show that for any minimal rational elliptic surface $X$ there is one such toric variety $Z_X$ and a divisor class ${\delta }_X\in \mathrm{Cl}(Z_X)$ such that the number of $(-1)$-curves of $X$ equals the dimension of the Riemann-Roch space of ${\delta }_X$. As an application we give the number of $(-1)$-curves of any such elliptic fibration of Halphen index $2$.

12 pages