Rational Nodal Curves with No Smooth Weierstrass Points

LetX denote the rational curve with n+1 nodes obtained from the Riemann sphere by identifying 0 with∞ and ζj with −ζj for j = 0, 1, . . . , n−1, where ζ is a primitive (2n)th root of unity. We show that if n is even, then X has no smooth Weierstrass points, while if n is odd, then X has 2n smooth Weierstrass points. C. Widland [14] showed that the rational curve with three nodes obtained from PC by identifying 0 with ∞, 1 with −1, and i with −i has no smooth Weierstrass points. This curve may be realized as the projective plane curve xy + yz = xz, which has “biflecnodes” at the points (1, 0, 0), (0, 1, 0), and (0, 0, 1), with each biflecnode being a Weierstrass point of weight 8. In this note, we show that this curve is one member of an infinite family of rational nodal curves with no smooth Weierstrass points. We remark that a general rational nodal curve with n nodes has n(n− 1) smooth Weierstrass points [9]. Let n be a positive integer. Let ζ denote a primitive (2n)th root of unity. Let X denote the rational curve with n + 1 nodes obtained from PC by identifying 0 with ∞, and ζ with −ζ for j = 0, 1, . . . , n− 1. Our main result is: (0.1) Theorem. (1) If n is even, then X has no smooth Weierstrass points. (2) If n is odd, then X has 2n smooth Weierstrass points at the (2n)th roots of −1. In the first section, we prove a theorem that can be used to show that fixed points of an automorphism are frequently Weierstrass points, which extends known results for smooth curves. This is then used in the second section to prove the main result. We thank Pierre Conner for very helpful discussions. 1 In this section, X and Y will denote integral projective curves over the complex numbers. If C is a curve, we let pa(C) denote the arithmetic genus of C and pg(C) denote the geometric genus of C. If P ∈ C, then we let ÕP denote the normalization Received by the editors September 14, 1994. 1991 Mathematics Subject Classification. Primary 14H55.