On the distribution of Weierstrass points on irreducible rational nodal curves (Q1175104)

Let \(X\) be an irreducible rational nodal curve of arithmetic genus \(g\geq2\), and let \({\mathcal L}\) be an invertible sheaf on \(X\) with \(\dim(H^ 0(X,{\mathcal L}))>0\) and \(\dim(H^ 1(X,{\mathcal L}))=0\). We denote by \(W({\mathcal L})\) the union of the sets of smooth Weierstrass points of \({\mathcal L}\), \(n\geq1\). The limit points of \(W({\mathcal L})\) always lie on a real one- dimensional subset of \(X\). In particular, \(W({\mathcal L})\) is not dense on \(X\). (In the classic case of smooth curves \textit{B. A. Olsen} [Ann. Math., II. Ser. 95, 357-364 (1972; Zbl 0232.14017)]\ had shown, that if \(\deg({\mathcal L})>0\), then \(W({\mathcal L})\) is dense in \(X\).) The main result of this paper is very interesting.