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I am asking mostly for reference if such a definition exists in the literature. I am also interested in the count if it appears somewhere.

Let $\mu:=(\mu_1,\ldots , \mu_n)\vdash d$ for positive integer $d$ then simple Hurwitz number $H_{g}(\mu)$ is defined to be the number of degree $d$ covering map say $f$ counted with weight $\frac{1}{m!\, Aut(f)}$ from $\bf{smooth} $ genus $g$ surface to $\mathbb{CP}^1$ having $m$ simple branch point and branch point of type $\mu$ at $\infty$.

$m2=g-2+n+d$ denote the number of simple singularities by Riemann Hurwitz formula.

Let me give a few counting of these numbers. $$H_0 (3)= \frac32$$ $$ H_1 (3)=\frac98$$ $$ H_0 (4)=\frac83$$

If there is in literature a definition of Hurwitz number that will extend the above definition of Hurwitz number from a nodal curve of genus $g$ to $CP^1$. Also, the weighted count would be finite. We need to take into account the nodal points.

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    $\begingroup$ The Hurwitz number in question can be described as the degree (suitably interpreted) of the map \pi: H -> M_{0,r}, where H parametrizes covers of the desired type and \pi remembers the marked target curve. One can compactify H to the space of admissible covers \bar{H} in the sense of Harris-Mumford, and by design there is a morphism \pi: \bar{H}\to \bar{M}_{0,n}. The global degree of this map should be the same as before, but one can then restrict to various boundary strata of \bar{M}_{0,n} and ask for the degrees there: it seems this should reduce to computing "smaller" Hurwitz numbers. $\endgroup$
    – Hans Sachs
    Dec 6, 2019 at 22:22
  • $\begingroup$ can you please give a concrete reference. I am not fluently in the terminology of the subject. $\endgroup$
    – GGT
    Dec 7, 2019 at 4:59
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    $\begingroup$ See section 4 of Harris-Mumford "On the Kodaira Dimension of the Moduli Space of Curves." I believe they only consider (degenerations of) simply branched covers of P^1, but one can easily carry over the definitions to more general ramification profiles. $\endgroup$
    – Hans Sachs
    Dec 7, 2019 at 17:06
  • $\begingroup$ citeseerx.ist.psu.edu/viewdoc/… So I guess you are referring to this file section. $\endgroup$
    – GGT
    Dec 8, 2019 at 12:39

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