On minimal rational elliptic surfaces

TL;DR

This paper constructs specific toric varieties to relate the number of (-1)-curves on minimal rational elliptic surfaces to the dimension of certain Riemann-Roch spaces, providing explicit counts for particular cases.

Contribution

It introduces a method to associate toric varieties with minimal rational elliptic surfaces, linking geometric curves to algebraic divisor class dimensions.

Findings

Number of (-1)-curves equals the dimension of a Riemann-Roch space for a related divisor class.

Constructs 13 projective Q-factorial Fano toric varieties.

Determines the count of (-1)-curves for elliptic fibrations with Halphen index 2.

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 $\delta_X\in {\rm Cl}(Z_X)$ 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 $2$.