Introducing Hurwitz Numbers for Severi-type Varieties
نویسنده
چکیده
Fixing an arbitrary point p ∈ CP and a triple (g, d, `) of nonnegative integers satisfying the inequality g ≤ (d+l−1 2 ) − (l 2 ) , we associate a natural Hurwitz number to the (open) Severi-type varietyWg,d,` consisting of all reduced irreducibke plane curves of degree d + l with genus g and having an ordinary singularity of order l at p (the remaining singular points of such curves being usual nodes). In the stable and semistable cases (d + l ≥ g + 2 and d + l ≥ g + 2, resp.) we calculate our Hurwitz number in terms of the appropriate classical (single) Hurwitz number. The most interesting unstable case (i.e. when d + 2l < g + 2) remains widely open.
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