Geometry & Topology

A desingularization of the main component of the moduli space of genus-one stable maps into $\mathbb P^n$

Ravi Vakil and Aleksey Zinger

Full-text: Open access

Abstract

We construct a natural smooth compactification of the space of smooth genus-one curves with k distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, that is, the space of stable genus-zero maps M̄0,k(n,d). In fact, our compactification is obtained from the singular space of stable genus-one maps M̄1,k(n,d) through a natural sequence of blowups along “bad” subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space. As a bonus, we obtain desingularizations of certain natural sheaves over the “main” irreducible component M̄1,k0(n,d) of M̄1,k(n,d). A number of applications of these desingularizations in enumerative geometry and Gromov–Witten theory are described in the introduction, including the second author’s proof of physicists’ predictions for genus-one Gromov–Witten invariants of a quintic threefold.

Article information

Source
Geom. Topol., Volume 12, Number 1 (2008), 1-95.

Dates
Received: 4 March 2007
Revised: 12 October 2007
Accepted: 8 October 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800013

Digital Object Identifier
doi:10.2140/gt.2008.12.1

Mathematical Reviews number (MathSciNet)
MR2377245

Zentralblatt MATH identifier
1134.14009

Subjects
Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}
Secondary: 53D99: None of the above, but in this section

Keywords
moduli space of stable maps genus one smooth compactification

Citation

Vakil, Ravi; Zinger, Aleksey. A desingularization of the main component of the moduli space of genus-one stable maps into $\mathbb P^n$. Geom. Topol. 12 (2008), no. 1, 1--95. doi:10.2140/gt.2008.12.1. https://projecteuclid.org/euclid.gt/1513800013


Export citation

References

  • M F Atiyah, R Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1–28
  • A Beauville, Quantum cohomology of complete intersections, Mat. Fiz. Anal. Geom. 2 (1995) 384–398
  • M Bershadsky, S Cecotti, H Ooguri, C Vafa, Holomorphic anomalies in topological field theories, Nuclear Phys. B 405 (1993) 279–304
  • A Bertram, Another way to enumerate rational curves with torus actions, Invent. Math. 142 (2000) 487–512
  • C Fontanari, Towards the cohomology of moduli spaces of higher genus stable maps
  • W Fulton, R Pandharipande, Notes on stable maps and quantum cohomology, from: “Algebraic geometry–-Santa Cruz 1995”, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI (1997) 45–96
  • A Gathmann, Absolute and relative Gromov–Witten invariants of very ample hypersurfaces, Duke Math. J. 115 (2002) 171–203
  • A Givental, The mirror formula for quintic threefolds, from: “Northern California Symplectic Geometry Seminar”, Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc., Providence, RI (1999) 49–62
  • M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
  • K Hori, S Katz, A Klemm, R Pandharipande, R Thomas, C Vafa, R Vakil, E Zaslow, Mirror symmetry, Clay Mathematics Monographs 1, American Mathematical Society, Providence, RI (2003) With a preface by Vafa
  • M Kontsevich, Y Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525–562
  • Y-P Lee, Quantum Lefschetz hyperplane theorem, Invent. Math. 145 (2001) 121–149
  • J Li, A Zinger, On the genus-one Gromov-Witten invariants of complete intersections
  • B H Lian, K Liu, S-T Yau, Mirror principle. I, Asian J. Math. 1 (1997) 729–763
  • D Maulik, R Pandharipande, A topological view of Gromov–Witten theory, Topology 45 (2006) 887–918
  • R Pandharipande, Intersections of $\mathbf{Q}$–divisors on Kontsevich's moduli space $\overline{M}_{0,n}(\mathbf{P}^r,d)$ and enumerative geometry, Trans. Amer. Math. Soc. 351 (1999) 1481–1505
  • Y Ruan, G Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995) 259–367
  • R Vakil, The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math. 529 (2000) 101–153
  • R Vakil, Murphy's law in algebraic geometry: badly-behaved deformation spaces, Invent. Math. 164 (2006) 569–590
  • R Vakil, A Zinger, A natural smooth compactification of the space of elliptic curves in projective space, Electron. Res. Announc. Amer. Math. Soc. 13 (2007) 53–59
  • A Zinger, Enumeration of genus-two curves with a fixed complex structure in $\mathbb P\sp 2$ and $\mathbb P\sp 3$, J. Differential Geom. 65 (2003) 341–467
  • A Zinger, Enumeration of one-nodal rational curves in projective spaces, Topology 43 (2004) 793–829
  • A Zinger, Counting rational curves of arbitrary shape in projective spaces, Geom. Topol. 9 (2005) 571–697
  • A Zinger, A sharp compactness theorem for genus-one pseudo-holomorphic maps
  • A Zinger, On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps, Adv. Math. 214 (2007) 878–933
  • A Zinger, Reduced genus-one Gromov-Witten invariants
  • A Zinger, Intersections of tautological classes on blowups of moduli spaces of genus-one curves, Michigan Math. J. 55 (2007) 535–560
  • A Zinger, The reduced genus-one Gromov-Witten invariants of Calabi-Yau hypersurfaces