Stability conditions on Fano threefolds of Picard number 1

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number $1$ with the exception of $12$ numerical cases.

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Bridgeland Stability Conditions on Fano Threefolds

Épijournal de Géométrie Algébrique ◽

10.46298/epiga.2017.volume1.2008 ◽

2017 ◽

Vol Volume 1 ◽

Author(s):

Marcello Bernardara ◽

Emanuele Macrì ◽

Benjamin Schmidt ◽

Xiaolei Zhao

Keyword(s):

Stability Conditions ◽

Fano Threefolds ◽

Frobenius Morphism ◽

Bridgeland Stability Conditions

We show the existence of Bridgeland stability conditions on all Fano threefolds, by proving a modified version of a conjecture by Bayer, Toda, and the second author. The key technical ingredient is a strong Bogomolov inequality, proved recently by Chunyi Li. Additionally, we prove the original conjecture for some toric threefolds by using the toric Frobenius morphism. Comment: 24 pages, 1 figure. Fifth version: Official version of the journal

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Fano threefolds with Picard number $2$ in positive characteristic

Kodai Mathematical Journal ◽

10.2996/kmj/1061901059 ◽

2003 ◽

Vol 26 (2) ◽

pp. 147-166 ◽

Cited By ~ 2

Author(s):

Natsuo Saito

Keyword(s):

Positive Characteristic ◽

Picard Number ◽

Fano Threefolds

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Towards the classification of weak Fano threefolds with ρ = 2

Open Mathematics ◽

10.2478/s11533-013-0261-5 ◽

2013 ◽

Vol 11 (9) ◽

Cited By ~ 2

Author(s):

Joseph Cutrone ◽

Nicholas Marshburn

Keyword(s):

Finite Number ◽

Type Ii ◽

Numerical Classification ◽

Picard Number ◽

Fano Threefolds ◽

Smooth Complex ◽

Complex Projective ◽

Extremal Ray ◽

Weak Fano

AbstractIn this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.

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ON THE PICARD NUMBER OF ALMOST FANO THREEFOLDS WITH PSEUDO-INDEX > 1

International Journal of Mathematics ◽

10.1142/s0129167x08004625 ◽

2008 ◽

Vol 19 (02) ◽

pp. 173-191 ◽

Cited By ~ 5

Author(s):

CINZIA CASAGRANDE ◽

PRISKA JAHNKE ◽

IVO RADLOFF

Keyword(s):

Picard Number ◽

Fano Threefolds ◽

Canonical Singularities ◽

Almost Fano

We study Gorenstein almost Fano threefolds X with canonical singularities and pseudoindex > 1. We show that the maximal Picard number of X is 10 in general, 3 if X is Fano, and 8 if X is toric. Moreover, we characterize the boundary cases. In the Fano case, we prove that the generalized Mukai conjecture holds.

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