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Weierstrass gap sequences at points of curves on some rational surfaces. (English) Zbl 1372.14028
Summary: Let \(\tilde{C}\) be a non-singular plane curve of degree \(d\geq 8\) with an involution \(\sigma \) over an algebraically closed field of characteristic 0 and \(\tilde{P}\) a point of \(\tilde{C}\) fixed by \(\sigma \). Let \(\pi : \tilde{C}\rightarrow C = \tilde{C} / \langle\sigma\rangle \) be the double covering. We set \(P = \pi (\tilde{P})\). When the intersection multiplicity at \(\tilde{P}\) of the curve \(\tilde{C}\) and the tangent line at \(\tilde{P}\) is equal to \(d - 3\) or \(d - 4\), we determine the Weierstrass gap sequence at \(P\) on \(C\) using blowing-ups and blowing-downs of some rational surfaces.
MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H50 Plane and space curves
14H30 Coverings of curves, fundamental group
14J26 Rational and ruled surfaces
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