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Loci of curves with subcanonical points in low genus. (English) Zbl 1364.14019
The aim of the paper is to compute the class, in their respective Chow rings, of some subschemes of \(\overline{\mathcal{M}_{3,1}}\) and \(\overline{\mathcal{M}_4}\) consisting of curves with subcanonical points (i.e. points \(p\) such that \((2g-2)p\) is a canonical divisor).
In particular, the classes of the closure of the following loci are explicitly computed in terms of \(\delta\), \(\lambda\) and \(\psi\)-classes: \[ \mathcal{H}yp_{3,1} = \{C\in \mathcal{M}_{3,1} \;| \;C \text{ is hyperelliptic with a marked Weierstrass point}\}, \] \[ \mathcal{F}_{3,1} = \{C\in \mathcal{M}_{3,1} \;| \;C \text{ is non-hyperelliptic with a marked hyperflex point}\}, \] and of a particular connected component of the locus \(\mathcal{H}_4\subseteq \mathcal{M}_4\) of curves with a subcanonical point.
As a corollary of their computations, the authors prove also that these subschemes are not complete intersections.
A particularly interesting statement, in my opinion, is Proposition 1.1, showing that some well-chosen classes form a basis of \(R^2(\overline{\mathcal{M}_{3,1}})\). The authors rely on the computations done in [E. Getzler, in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30–July 4, 1997, and in Kyoto, Japan, July 7–11 1997. Singapore: World Scientific. 73–106 (1998; Zbl 1021.81056)] to obtain the result.
The proofs have the character of careful computations, in which the result of each passage is explicitly written down leaving no gaps in the argument. In the places where some known result is used, precise references are given.

MSC:
14H10 Families, moduli of curves (algebraic)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C15 (Equivariant) Chow groups and rings; motives
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H45 Special algebraic curves and curves of low genus
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