Weierstrass points on trigonal curves. I: The ramification points (Q910448)

Let C be a smooth curve of genus \(g\geq 5\) and assume that P is a Weierstrass point on C with first non-gap equal to 3. The gap sequence at P can be written \((1, 2, 4, 5, \dots, 3n-2, 3n-1, 3n+\epsilon, 3n+3+\epsilon, \ldots, 3(g-n-1)+\epsilon)\) and is completely determined by the numbers n and \(\epsilon\) satisfying \((g-1)/3\leq n\leq g/2\) and \(1\leq \epsilon \leq 2\), \(\epsilon\in {\mathbb{Z}}\). In this notation P is called of the n-th kind and of type I (respectively type II) if \(\epsilon =1\) (respectively 2). On a curve of genus \(g\geq 5\) all Weierstrass points with first non-gap equal to 3 are of the same kind. For n, t, t(II) possible values let \(M_{g,3,n}(t,t(II))\) be the locus in the coarse moduli space of curves of genus g of the curves C that have t Weierstrass points with first non-gap equal to 3 of kind n, t(II) of them being of type II. The author computes \(\dim (M_{g,3,n}(t,t(II)))\) and in the cases \(t=1\) he proves that \(M_{g,3,n}(t,t(II))\) is irreducible and also for C a general point in \(M_{g,3,n}(t,t(II)\) and \(C\to {\mathbb{P}}_ 1\) a trigonal covering he gives the gap sequences at the ordinary ramification points and proves that the other Weierstrass points are normal.