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Examples of Fano varieties of index one that are not birationally rigid
Author:
Ana-Maria Castravet
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3783-3788
MSC (2000):
Primary 14E07; Secondary 14H60
Posted:
September 12, 2007
MathSciNet review:
2341927
Full-text PDF Free Access
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Abstract: A conjecture of Pukhlikov states that a smooth Fano variety of dimension at least 4 and index one is birationally rigid. We show that a general member of the linear system given by the ample generator of the Picard group of the moduli space of stable, rank 2 bundles with fixed determinant of odd degree on a curve of genus at least 3 is not birationally rigid.
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Aaron
Bertram, Moduli of rank-2 vector bundles, theta divisors, and the
geometry of curves in projective space, J. Differential Geom.
35 (1992), no. 2, 429–469. MR 1158344
(93g:14037)
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Sonia
Brivio and Alessandro
Verra, The theta divisor of
𝑆𝑈_{𝐶}(2,2𝑑)^{𝑠} is very ample if
𝐶 is not hyperelliptic, Duke Math. J. 82
(1996), no. 3, 503–552. MR 1387683
(97e:14017), http://dx.doi.org/10.1215/S0012-7094-96-08222-8
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Castravet, Rational families of vector bundles on curves,
Internat. J. Math. 15 (2004), no. 1, 13–45. MR 2039210
(2005i:14038), http://dx.doi.org/10.1142/S0129167X0400220X
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Alessio
Corti, Singularities of linear systems and 3-fold birational
geometry, Explicit birational geometry of 3-folds, London Math. Soc.
Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000,
pp. 259–312. MR 1798984
(2001k:14041)
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de Fernex, T., Adjunction beyond thresholds and birationally rigid hypersurfaces; arxiv:math. AG/0604213
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Tommaso
de Fernex, Lawrence
Ein, and Mircea
Mustaţă, Bounds for log canonical thresholds with
applications to birational rigidity, Math. Res. Lett.
10 (2003), no. 2-3, 219–236. MR 1981899
(2004e:14060)
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J.-M.
Drezet and M.
S. Narasimhan, Groupe de Picard des variétés de
modules de fibrés semi-stables sur les courbes
algébriques, Invent. Math. 97 (1989),
no. 1, 53–94 (French). MR 999313
(90d:14008), http://dx.doi.org/10.1007/BF01850655
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Iskovskikh, V.A., Manin Ju. I., Three-dimensional quartics and counterexamples to the Luroth problem, Math. USSR-Sb., 15 (1971), 141-166
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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], vol. 32,
Springer-Verlag, Berlin, 1996. MR 1440180
(98c:14001)
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Kollár, Yoichi
Miyaoka, and Shigefumi
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1 (1992), no. 3, 429–448. MR 1158625
(93i:14014)
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E. Newstead, Characteristic classes of stable
bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337–345. MR 0316452
(47 #4999), http://dx.doi.org/10.1090/S0002-9947-1972-0316452-9
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S.
Ramanan, The moduli spaces of vector bundles over an algebraic
curve, Math. Ann. 200 (1973), 69–84. MR 0325615
(48 #3962)
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Aleksandr
V. Pukhlikov, Birational automorphisms of Fano hypersurfaces,
Invent. Math. 134 (1998), no. 2, 401–426. MR 1650332
(99i:14046), http://dx.doi.org/10.1007/s002220050269
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A.
V. Pukhlikov, Birationally rigid double Fano hypersurfaces,
Mat. Sb. 191 (2000), no. 6, 101–126 (Russian,
with Russian summary); English transl., Sb. Math. 191
(2000), no. 5-6, 883–908. MR 1777571
(2001h:14054), http://dx.doi.org/10.1070/SM2000v191n06ABEH000485
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A.
V. Pukhlikov, Birationally rigid Fano complete intersections,
J. Reine Angew. Math. 541 (2001), 55–79. MR 1876285
(2003a:14015), http://dx.doi.org/10.1515/crll.2001.095
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Aleksandr
V. Pukhlikov, Birationally rigid Fano varieties, The Fano
Conference, Univ. Torino, Turin, 2004, pp. 659–681. MR 2112597
(2005j:14017)
- [B]
- Bertram, A., Moduli of rank
 vector bundles, theta divisors, and the geometry of curves in projective space, J. Differential Geometry, 35, (1992), 429-469 MR 1158344 (93g:14037)
- [BV]
- Brivio, S., Verra, A., The theta divisor of
 is very ample if  is not hyperelliptic, Duke Math. J., 82, (1996), No. 3, 503-552 MR 1387683 (97e:14017)
- [Ca]
- Castravet, A.-M., Rational families of vector bundles on curves, International Journal of Mathematics, 15, No. 1 (2004); arxiv:math. AG/0302133 and math. AG/0302135 MR 2039210 (2005i:14038)
- [Co]
- Corti, A., Singularities of linear systems and
 -fold birational geometry, in Explicit birational geometry of  -folds, 259-312, Cambridge Univ. Press, Cambridge, 2000 MR 1798984 (2001k:14041)
- [dF]
- de Fernex, T., Adjunction beyond thresholds and birationally rigid hypersurfaces; arxiv:math. AG/0604213
- [dFEM]
- de Fernex, T, Ein, L., Mustata, M., Bounds on log-canonical thresholds with application to birational rigidity, Math. Res. Lett., 10, (2003), 219-236 MR 1981899 (2004e:14060)
- [DN]
- Drézét, J.-M., Narasimhan, M.S., Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math., 97, (1989), 53-94 MR 999313 (90d:14008)
- [IM]
- Iskovskikh, V.A., Manin Ju. I., Three-dimensional quartics and counterexamples to the Luroth problem, Math. USSR-Sb., 15 (1971), 141-166
- [K]
- Kollár, J.,Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge., 32, Springer-Verlag, Berlin, 1996 MR 1440180 (98c:14001)
- [KMM]
- Kollár, J., Miyaoka, Y., Mori, S., Rationally Connected Varieties , J. Alg. Geom., I (1992), 429-448 MR 1158625 (93i:14014)
- [N]
- Newstead, P.E., Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc., 169, (1972), 337-345 MR 0316452 (47:4999)
- [R]
- Ramanan, S., The moduli spaces of vector bundles over an algebraic curve, Math. Ann., 200, (1973), 69-84 MR 0325615 (48:3962)
- [P1]
- Pukhlikov, A.V., Birational automorphisms of Fano hypersurfaces, Invent. Math., 134, (1998), no. 2, 401-426 MR 1650332 (99i:14046)
- [P2]
- Pukhlikov, A.V., Birationally rigid Fano double hypersurfaces, Sbornik: Mathematics, 191, (2000), no. 6, 101-126 MR 1777571 (2001h:14054)
- [P3]
- Pukhlikov, A.V., Birationally rigid Fano complete intersections, Crelle J. für die reine und angew. Math, 541, (2001), 55-79 MR 1876285 (2003a:14015)
- [P4]
- Pukhlikov, A.V., Birationally rigid Fano varieties, Proceedings of Fano Conference, 659-681, Univ. Torino, Turin, 2004; arXiv:math.AG/0310267 MR 2112597 (2005j:14017)
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Additional Information
Ana-Maria Castravet
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Address at time of publication:
Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
Email:
noni@math.utexas.edu, noni@math.umass.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08948-4
PII:
S 0002-9939(07)08948-4
Received by editor(s):
May 5, 2006
Received by editor(s) in revised form:
September 17, 2006
Posted:
September 12, 2007
Communicated by:
Ted Chinburg
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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