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The blow-up of the anticanonical base point on a del Pezzo surface S of degree 1 gives rise to a rational elliptic surface 
$\mathscr {E}$ with only irreducible fibers. The sections of minimal height of 
$\mathscr {E}$ are in correspondence with the 
$240$ exceptional curves on S. A natural question arises when studying the configuration of these curves: if a point on S is contained in “many” exceptional curves, is it torsion on its fiber on 
$\mathscr {E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree 
$2$, where there are 56 exceptional curves, that if “many” equals 
$4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if ‘many’ equals 
$9$ or more. Moreover, we give counterexamples where a non-torsion point lies in the intersection of 
$7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.
