## Algebra & Number Theory

### Extremality of loci of hyperelliptic curves with marked Weierstrass points

#### Abstract

The locus of genus-two curves with $n$ marked Weierstrass points has codimension $n$ inside the moduli space of genus-two curves with $n$ marked points, for $n ≤ 6$. It is well known that the class of the closure of the divisor obtained for $n = 1$ spans an extremal ray of the cone of effective divisor classes. We generalize this result for all $n$: we show that the class of the closure of the locus of genus-two curves with $n$ marked Weierstrass points spans an extremal ray of the cone of effective classes of codimension $n$, for $n ≤ 6$. A related construction produces extremal nef curve classes in moduli spaces of pointed elliptic curves.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 9 (2016), 1935-1948.

Dates
Revised: 17 June 2016
Accepted: 10 September 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842609

Digital Object Identifier
doi:10.2140/ant.2016.10.1935

Mathematical Reviews number (MathSciNet)
MR3576115

Zentralblatt MATH identifier
1354.14048

Subjects
Primary: 14H99: None of the above, but in this section
Secondary: 14C99: None of the above, but in this section

#### Citation

Chen, Dawei; Tarasca, Nicola. Extremality of loci of hyperelliptic curves with marked Weierstrass points. Algebra Number Theory 10 (2016), no. 9, 1935--1948. doi:10.2140/ant.2016.10.1935. https://projecteuclid.org/euclid.ant/1510842609

#### References

• P. Belorousski and R. Pandharipande, “A descendent relation in genus 2”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 29:1 (2000), 171–191.
• J. Bergstr öm, “Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves”, Doc. Math. 14 (2009), 259–296.
• D. Chen and I. Coskun, “Extremal effective divisors on $\overline{\mathscr{M}}\sb {1,n}$”, Math. Ann. 359:3-4 (2014), 891–908.
• D. Chen and I. Coskun, “Extremal higher codimension cycles on moduli spaces of curves”, Proc. Lond. Math. Soc. $(3)$ 111:1 (2015), 181–204.
• D. Eisenbud and J. Harris, “The Kodaira dimension of the moduli space of curves of genus $\geq 23$”, Invent. Math. 90:2 (1987), 359–387.
• C. Faber, “A conjectural description of the tautological ring of the moduli space of curves”, pp. 109–129 in Moduli of curves and abelian varieties, edited by C. Faber and E. Looijenga, Aspects Math. E33, Vieweg, Braunschweig, 1999.
• C. Faber and R. Pandharipande, “Relative maps and tautological classes”, J. Eur. Math. Soc. $($JEMS$)$ 7:1 (2005), 13–49.
• W. Fulton and R. MacPherson, “A compactification of configuration spaces”, Ann. of Math. $(2)$ 139:1 (1994), 183–225.
• E. Getzler, “Topological recursion relations in genus $2$”, pp. 73–106 in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), edited by M.-H. Saito et al., World Sci. Publ., River Edge, NJ, 1998.
• T. Graber and R. Vakil, “Relative virtual localization and vanishing of tautological classes on moduli spaces of curves”, Duke Math. J. 130:1 (2005), 1–37.
• E. Looijenga, “On the tautological ring of ${\mathscr M}\sb g$”, Invent. Math. 121:2 (1995), 411–419.
• D. Petersen, “Tautological rings of spaces of pointed genus two curves of compact type”, Compos. Math. 152:7 (2016), 1398–1420.
• D. Petersen and O. Tommasi, “The Gorenstein conjecture fails for the tautological ring of $\overline{\cal M}\sb {2,n}$”, Invent. Math. 196:1 (2014), 139–161.
• W. F. Rulla, The birational geometry of moduli space M(3) and moduli space M(2,1), Ph.D. thesis, University of Texas at Austin, 2001, hook http://search.proquest.com/docview/304719193 \posturlhook.
• W. F. Rulla, “Effective cones of quotients of moduli spaces of stable $n$-pointed curves of genus zero”, Trans. Amer. Math. Soc. 358:7 (2006), 3219–3237.
• N. Tarasca, “Double total ramifications for curves of genus 2”, Int. Math. Res. Not. 2015:19 (2015), 9569–9593.