Algebra & Number Theory

Moduli of morphisms of logarithmic schemes

Jonathan Wise

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show that there is a logarithmic algebraic space parameterizing logarithmic morphisms between fixed logarithmic schemes when those logarithmic schemes satisfy natural hypotheses. As a corollary, we obtain the representability of the stack of stable logarithmic maps from logarithmic curves to a fixed target without restriction on the logarithmic structure of the target.

An intermediate step requires a left adjoint to pullback of étale sheaves, whose construction appears to be new in the generality considered here, and which may be of independent interest.

Article information

Source
Algebra Number Theory, Volume 10, Number 4 (2016), 695-735.

Dates
Received: 14 September 2014
Revised: 28 October 2015
Accepted: 22 March 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842512

Digital Object Identifier
doi:10.2140/ant.2016.10.695

Mathematical Reviews number (MathSciNet)
MR3519093

Zentralblatt MATH identifier
1343.14020

Subjects
Primary: 14H10: Families, moduli (algebraic)
Secondary: 14D23: Stacks and moduli problems 14A20: Generalizations (algebraic spaces, stacks)

Keywords
logarithmic geometry moduli

Citation

Wise, Jonathan. Moduli of morphisms of logarithmic schemes. Algebra Number Theory 10 (2016), no. 4, 695--735. doi:10.2140/ant.2016.10.695. https://projecteuclid.org/euclid.ant/1510842512


Export citation

References

  • D. Abramovich and Q. Chen, “Stable logarithmic maps to Deligne–Faltings pairs II”, Asian J. Math. 18:3 (2014), 465–488.
  • D. Abramovich, Q. Chen, W. D. Gillam, and S. Marcus, “The evaluation space of logarithmic stable maps”, preprint, 2010.
  • K. Ascher and S. Molcho, “Logarithmic stable toric varieties and their moduli”, preprint, 2015.
  • Q. Chen, “Stable logarithmic maps to Deligne–Faltings pairs I”, Ann. of Math. $(2)$ 180:2 (2014), 455–521.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III”, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255.
  • W. D. Gillam, “Logarithmic stacks and minimality”, Internat. J. Math. 23:7 (2012), 1250069, 38.
  • W. D. Gillam and S. Molcho, “Stable log maps as moduli of flow lines”, preprint, 2013, hook http://math.colorado.edu/~samo2465/flows3.pdf \posturlhook.
  • M. Gross and B. Siebert, “Logarithmic Gromov–Witten invariants”, J. Amer. Math. Soc. 26:2 (2013), 451–510.
  • J. Hall and D. Rydh, “Coherent Tannaka duality and algebraicity of Hom-stacks”, preprint, 2015.
  • K. Kato, “Logarithmic structures of Fontaine–Illusie”, pp. 191–224 in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), edited by J.-I. Igusa, Johns Hopkins Univ. Press, Baltimore, MD, 1989.
  • G. Laumon and L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 39, Springer, Berlin, 2000.
  • S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics 5, Springer, New York, 1998.
  • M. C. Olsson, “Logarithmic geometry and algebraic stacks”, Ann. Sci. École Norm. Sup. $(4)$ 36:5 (2003), 747–791.
  • M. C. Olsson, Compactifying moduli spaces for abelian varieties, Lecture Notes in Mathematics 1958, Springer, Berlin, 2008.
  • M. Romagny, “Composantes connexes et irréductibles en familles”, Manuscripta Math. 136:1-2 (2011), 1–32.
  • J. C. Rosales and P. A. García-Sánchez, Finitely generated commutative monoids, Nova Science Publishers, Commack, NY, 1999.