The blow up of the anticanonical base point on X, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface E with only irreducible fibers. The sections of minimal height of E are in correspondence with the 240 exceptional curves on X. A natural question arises when studying the configuration of those curves : 

If a point of X is contained in « many » exceptional curves, it is torsion on its fiber on E?

In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there is 56 exceptional curves) that if « many » equals 4 or more, then yes. With Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if « many » equals 9 or more, then yes. Additionnally we find counterexamples where a torsion point lies at the intersection lies at the intersection of 7 exceptional curves.

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