On higher Weierstrass points
Let \(C_ g\) be the coarse moduli space of genus g of pointed curves over the complex numbers and \(W^ n_ g=\{(X,p)\in C_ g\); is an n- Weierstrass point of \(X\}\). - The main result is the following theorem: (i) \(W^ n_ g\) is irreducible for \(g\geq 4\), \(n\geq 1\) and for \(g=3\), \(n\neq 2.\) (ii) \(W^ 2_ 3=W^ 1_ 3+S\) where S is an irreducible divisor. Among the many interesting results of this paper I should mention: (a) A new proof of the following theorem of Lax: If \(g\geq 3\) then a general smooth curve of genus g has only ordinary n-Weierstrass points. (b) An n- Weierstrass point p is said to be dimensionally proper if the locus of n- Weierstrass points near p in the versal deformation family has codimension equal to the weight at p. The authors prove the following existence theorem for n-Weierstrass points: Let \(\beta =(\beta_ 0,...,\beta_ r)\) where \(r=(2n-1)(g-1)-1\) with \(0\leq \beta_ i\leq \beta_{i+1}\); let \(w(\beta):=\sum^{r}_{i=0}\beta_ i \). For \(n\geq 3\) and \(g\geq 2\) (or for \(n=2\) and \(g\geq 3)\) there exists a dimensionally proper n-Weierstrass point on some smooth curve of genus \(g\) having \(\beta\) as its ramification sequence, if \(w(\beta)\leq g-1\) (or respectively \(w(\beta)\leq g-2)\). (c) The authors compute the linear equivalence class of the closure of \(W^ n_ g\) in the moduli space of stable pointed curves.
scientific article; zbMATH DE number 4202470
scientific article