The Michigan Mathematical Journal

A Note on Rational Curves on General Fano Hypersurfaces

Dennis Tseng

Advance publication

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Abstract

We show that the Kontsevich space of rational curves of degree at most roughly 222n on a general hypersurface XPn of degree n1 is equidimensional of expected dimension and has two components: one consisting generically of smooth embedded rational curves and the other consisting of multiple covers of a line. This proves more cases of a conjecture of Coskun, Harris, and Starr and shows that the Gromov–Witten invariants in these cases are enumerative.

Article information

Source
Michigan Math. J., Advance publication (2019), 20 pages.

Dates
Received: 8 November 2017
Revised: 1 October 2018
First available in Project Euclid: 6 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1567735281

Digital Object Identifier
doi:10.1307/mmj/1567735281

Subjects
Primary: 14H10: Families, moduli (algebraic) 14J45: Fano varieties 14J70: Hypersurfaces

Citation

Tseng, Dennis. A Note on Rational Curves on General Fano Hypersurfaces. Michigan Math. J., advance publication, 6 September 2019. doi:10.1307/mmj/1567735281. https://projecteuclid.org/euclid.mmj/1567735281


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References

  • [1] R. Beheshti and N. Mohan Kumar, Spaces of rational curves on complete intersections, Compos. Math. 149 (2013), no. 6, 1041–1060.
  • [2] K. Behrend and Y. Manin, Stacks of stable maps and Gromov–Witten invariants, Duke Math. J. 85 (1996), no. 1, 1–60.
  • [3] T. Browning and P. Vishe, Rational curves on smooth hypersurfaces of low degree, Algebra Number Theory 11 (2017), no. 7, 1657–1675.
  • [4] A.-M. Castravet, Rational families of vector bundles on curves, Internat. J. Math. 15 (2004), no. 1, 13–45.
  • [5] I. Coskun and J. Starr, Rational curves on smooth cubic hypersurfaces, Int. Math. Res. Not. IMRN 24 (2009), 4626–4641.
  • [6] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., 62, pp. 45–96, Amer. Math. Soc., Providence, RI, 1997.
  • [7] K. Furukawa, Rational curves on hypersurfaces, J. Reine Angew. Math. 665 (2012), 157–188.
  • [8] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Publ. Math. Inst. Hautes Études Sci. 20 (1964), 259.
  • [9] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 24 (1965), 231.
  • [10] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci. 28 (1966), 255.
  • [11] L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), no. 3, 491–506.
  • [12] J. Harris, Algebraic geometry, Grad. Texts in Math., 133, Springer-Verlag, New York, 1992, a first course.
  • [13] J. Harris, M. Roth, and J. Starr, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106.
  • [14] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror symmetry, Clay Math. Monogr., 1, American Mathematical Society, Providence, RI, 2003, with a preface by Vafa.
  • [15] J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 32, Springer-Verlag, Berlin, 1996.
  • [16] C. Minoccheri, On the arithmetic of weighted complete intersections of low degree, preprint, arXiv:1608.01703v1.
  • [17] E. Riedl and D. Yang, Kontsevich spaces of rational curves on Fano hypersurfaces, Journal für die reine und angewandte Mathematik (Crelles Journal), forthcoming.
  • [18] K. Slavov, The moduli space of hypersurfaces whose singular locus has high dimension, Math. Z. 279 (2015), no. 1–2, 139–162.
  • [19] The Stacks Project Authors. Stacks project, http://stacks.math.columbia.edu, 2017.
  • [20] D. Tseng, Collections of hypersurfaces containing a curve, International Mathematics Research Notices, p. rny120, 2018.