# Generalized Mukai conjecture for special Fano varieties

Marco Andreatta; Elena Chierici; Gianluca Occhetta

Open Mathematics (2004)

- Volume: 2, Issue: 2, page 272-293
- ISSN: 2391-5455

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topMarco Andreatta, Elena Chierici, and Gianluca Occhetta. "Generalized Mukai conjecture for special Fano varieties." Open Mathematics 2.2 (2004): 272-293. <http://eudml.org/doc/268759>.

@article{MarcoAndreatta2004,

abstract = {Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.},

author = {Marco Andreatta, Elena Chierici, Gianluca Occhetta},

journal = {Open Mathematics},

keywords = {14J45; 14E30},

language = {eng},

number = {2},

pages = {272-293},

title = {Generalized Mukai conjecture for special Fano varieties},

url = {http://eudml.org/doc/268759},

volume = {2},

year = {2004},

}

TY - JOUR

AU - Marco Andreatta

AU - Elena Chierici

AU - Gianluca Occhetta

TI - Generalized Mukai conjecture for special Fano varieties

JO - Open Mathematics

PY - 2004

VL - 2

IS - 2

SP - 272

EP - 293

AB - Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.

LA - eng

KW - 14J45; 14E30

UR - http://eudml.org/doc/268759

ER -

## References

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- [8] J. Kollár: Rational Curves on Algebraic Varieties, Ergebnisse der Math. Vol. 32, Springer-Verlag, 1996.
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- [11] S. Mukai: “Open problems”, In: Birational geometry of algebraic varieties, Taniguchi Foundation, Katata, 1988.
- [12] G. Occhetta: A characterization of products of projective spaces, preprint, February 2003, http://www.science.unitn.it/∼occhetta.
- [13] J.A. Wiśniewski: “On a conjecture of Mukai”, Manuscripta Math., Vol. 68, (1990), pp. 135–141. Zbl0715.14033

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