On the Kodaira dimension of the moduli space of nodal curves. (English) Zbl 1473.14051
The author shows that the moduli space of geometric genus \(g\) curves with \(n\) nodes is of general type for \(g\geq 24\) and for all \(n\). For low genus \(5\leq g \leq 23\), the author also describes explicit bounds on \(n\) for which the general-type result holds. For \(g\geq 24\), the result essentially relies on the fact that the moduli space of genus \(g\) curves is of general type in the same range. However for low genus, the author has to work with a number of specific effective divisors of Brill-Noether type and of Weierstrass type, in order to decompose the canonical class of the moduli space as ample plus effective.
Reviewer: Dawei Chen (Chestnut Hill)
MSC:
| 14H10 | Families, moduli of curves (algebraic) |
| 14H51 | Special divisors on curves (gonality, Brill-Noether theory) |
| 14D22 | Fine and coarse moduli spaces |
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