# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves. Vol. I, volume 267 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, New York (1985) · Zbl 0559.14017 [2] Bergström, J, Cohomology of moduli spaces of curves of genus three via point counts, J. Reine Angew. Math., 622, 155-187, (2008) · Zbl 1158.14025 [3] Bainbridge, M; Möller, M, The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3, Acta Math., 208, 1-92, (2012) · Zbl 1250.14014 [4] Chen, D; Coskun, I, Extremal higher codimension cycles on moduli spaces of curves, Proc. Lond. Math. Soc., 111, 181-204, (2015) · Zbl 1357.14039 [5] Chen, D, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228, 1135-1162, (2011) · Zbl 1227.14030 [6] Chen, D, Strata of abelian differentials and the Teichmüller dynamics, J. Mod. Dyn., 7, 135-152, (2013) · Zbl 1273.14054 [7] Chen, D; Möller, M, Nonvarying sums of Lyapunov exponents of abelian differentials in low genus, Geom. Topol., 16, 2427-2479, (2012) · Zbl 1266.14018 [8] Cornalba, M.: Moduli of curves and theta-characteristics. In: Lectures on Riemann Surfaces (Trieste, 1987), pp. 560-589. World Sci. Publ., Teaneck (1989) · Zbl 0800.14011 [9] Caporaso, L; Sernesi, E, Recovering plane curves from their bitangents, J. Algebra Geom., 12, 225-244, (2003) · Zbl 1080.14523 [10] Cukierman, F, Families of Weierstrass points, Duke Math. J., 58, 317-346, (1989) · Zbl 0687.14026 [11] Diaz, S, Exceptional Weierstrass points and the divisor on moduli space that they define, Mem. Amer. Math. Soc, 56, iv+69, (1985) · Zbl 0581.14018 [12] Dolgachev, I; Kanev, V, Polar covariants of plane cubics and quartics, Adv. Math., 98, 216-301, (1993) · Zbl 0791.14013 [13] Eisenbud, D; Harris, J, Irreducibility of some families of linear series with brill-Noether number $$-1$$, Ann. Sci. École Norm. Sup. (4), 22, 33-53, (1989) · Zbl 0691.14006 [14] Faber, C, Chow rings of moduli spaces of curves. I. the Chow ring of $$\overline{\cal M}_3$$, Ann. Math. (2), 132, 331-419, (1990) · Zbl 0721.14013 [15] Faber, C, Chow rings of moduli spaces of curves. II. some results on the Chow ring of $$\overline{\cal M}_4$$, Ann. Math. (2), 132, 421-449, (1990) · Zbl 0735.14021 [16] Farkas, G, The birational type of the moduli space of even spin curves, Adv. Math., 223, 433-443, (2010) · Zbl 1183.14020 [17] Farkas, G.: Brill-Noether geometry on moduli spaces of spin curves. In: Classification of Algebraic Varieties, EMS Ser. Congr. Rep., pp. 259-276. Eur. Math. Soc., Zürich (2011) · Zbl 1223.14028 [18] Faber, C; Pandharipande, R, Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS), 7, 13-49, (2005) · Zbl 1084.14054 [19] Faber, C., Pagani, N.: The class of the bielliptic locus in genus 3. Preprint, arXiv:1206.4301. To appear in Int. Math. Res. Not. (2015) · Zbl 1329.14061 [20] Farkas, G; Verra, A, The geometry of the moduli space of odd spin curves, Ann. Math. (2), 180, 927-970, (2014) · Zbl 1325.14045 [21] Getzler, E.: Topological recursion relations in genus 2. In: Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), pp. 73-106. World Sci. Publ., River Edge (1998) · Zbl 1021.81056 [22] Getzler, E., Looijenga, E.: The Hodge polynomial of $$\overline{\cal M}_{3,1}$$. Preprint, arXiv:math/9910174 (1999) · Zbl 0691.14006 [23] Hain, R.: Normal functions and the geometry of moduli spaces of curves. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli, vol. I of Advanced Lectures in Mathematics. Somerville: International Press (2013) · Zbl 1322.14049 [24] Harris, J; Mumford, D, On the Kodaira dimension of the moduli space of curves, Invent. Math., 67, 23-88, (1982) · Zbl 0506.14016 [25] Jensen, D, Birational contractions of $$\overline{M}_{3,1}$$ and $$\overline{M}_{4,1}$$, Trans. Amer. Math. Soc., 365, 2863-2879, (2013) · Zbl 1345.14035 [26] Kontsevich, M; Zorich, A, Connected components of the moduli spaces of abelian differentials with prescribed singularities, Invent. Math., 153, 631-678, (2003) · Zbl 1087.32010 [27] Logan, A, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math., 125, 105-138, (2003) · Zbl 1066.14030 [28] Scorza, G, Sopra le curve canoniche di uno spazio lineare qualunque e sopra certi loro covarianti quartici, Atti R. Accad. Sci. Torino, 35, 765-773, (1900) [29] Tarasca, N, Brill-Noether loci in codimension two, Compos. Math., 149, 1535-1568, (2013) · Zbl 1297.14030 [30] Tarasca, N, Double total ramifications for curves of genus 2, Int. Math. Res. Not., 19, 9569-9593, (2015) · Zbl 1348.14077 [31] Montserrat, TiB, The divisor of curves with a vanishing theta-null, Compos. Math., 66, 15-22, (1988) · Zbl 0663.14017 [32] Yang, S.: Calculating intersection numbers on moduli spaces of curves. Preprint, arXiv:0808.1974v2 (2010) · Zbl 1345.14035 [33] Zorich, A.: Flat surfaces. In: Frontiers in Number Theory, Physics, and Geometry. I, pp. 437-583. Springer, Berlin (2006) · Zbl 1129.32012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.