The irreducibility of the spaces of rational curves on del Pezzo surfaces
Author:
Damiano Testa
Journal:
J. Algebraic Geom. 18 (2009), 37-61
DOI:
https://doi.org/10.1090/S1056-3911-08-00484-0
Published electronically:
August 14, 2008
MathSciNet review:
2448278
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Abstract |
References |
Additional Information
Abstract: Let $X$ be a del Pezzo surface of degree $d \geq 2$. We prove that the spaces $\mathcal M_{0,0} \bigl ( X, \beta \bigr )$ are either empty or irreducible.
References
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References
- R. Beheshti, J. Starr, Rational surfaces in index-one Fano hypersurfaces, J. Algebraic Geom. 17 (2008), no. 2, 255–276. MR 2369086
- A. J. de Jong, J. Starr, Cubic fourfolds and spaces of rational curves, Illinois J. Math. 48 (2004), no. 2, 415-450. MR 2085418 (2006e:14007)
- A. J. de Jong, J. Starr, Higher Fano manifolds and rational surfaces, math.AG/0602647.
- P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. No. 36 (1969) 75-109. MR 0262240 (41:6850)
- W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. 90 (1969) 542-575. MR 0260752 (41:5375)
- T. Graber, J. Harris, J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57-67. MR 1937199 (2003m:14081)
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. No. 32 (1967). MR 0238860 (39:220)
- J. Harris, On the Severi problem, Invent. Math. 84 (1986), 445-461. MR 837522 (87f:14012)
- J. Harris, M. Roth, J. Starr, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73-106. MR 2070144 (2005e:14067)
- J. Harris, J. Starr, Rational curves on hypersurfaces of low degree II, Compos. Math. 141 (2005), no. 1, 35-92. MR 2099769 (2006c:14039)
- R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57:3116)
- B. Kim, R. Pandharipande, The connectedness of the moduli space of maps to homogeneous spaces, Symplectic geometry and mirror symmetry (Seoul, 2000), 187-201, World Sci. Publ., River Edge, NJ, 2001. MR 1882330 (2002k:14021)
- J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 32, Springer-Verlag, Berlin, 1996. MR 1440180 (98c:14001)
- R. Lazarsfeld, Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 48, Springer-Verlag, Berlin, 2004. MR 2095471 (2005k:14001a)
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- F. Severi, Vorlesungen über Algebraische Geometrie, Anhang F, Leipzig, Teubner, 1921.
Additional Information
Damiano Testa
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication:
Università di Roma “La Sapienza” 00185 Roma, Italy
MR Author ID:
794104
Email:
testa@mat.uniroma1.it
Received by editor(s):
October 6, 2006
Received by editor(s) in revised form:
February 21, 2007
Published electronically:
August 14, 2008
