scholarly journals On the existence of certain weak Fano threefolds of Picard number two

2017 ◽  
Vol 120 (1) ◽  
pp. 68 ◽  
Author(s):  
Maxim Arap ◽  
Joseph Cutrone ◽  
Nicholas Marshburn
Keyword(s):  
Picard Number ◽  
Fano Threefolds ◽  
Canonical Map ◽  
Blowing Up ◽  
Numerical Invariants ◽  
Weak Fano

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.

Towards the classification of weak Fano threefolds with ρ = 2

2013 ◽  
Vol 11 (9) ◽  
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.

scholarly journals K3 surfaces with non-symplectic involution and compact irreducible G2-manifolds

2011 ◽  
Vol 151 (2) ◽  
pp. 193-218 ◽  
Author(s):  
ALEXEI KOVALEV ◽  
NAM-HOON LEE
Keyword(s):  
Betti Numbers ◽  
Matching Condition ◽  
K3 Surfaces ◽  
Connected Sum ◽  
Fano Threefolds ◽  
New Class ◽  
Blowing Up ◽  
G2 Manifolds

AbstractWe consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G2 developed by the first named author. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors and the latter K3 surfaces should satisfy a certain ‘matching condition’ intertwining on their periods and Kähler classes. Suitable examples of threefolds were previously obtained by blowing up curves in Fano threefolds.In this paper, we give a large new class of suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds as above, and admit matching pairs leading to topologically new examples of compact irreducible G2-manifolds. ‘Geography’ of the values of Betti numbers b2, b3 for the new (and previously known) examples of irreducible G2 manifolds is also discussed.

scholarly journals BOUNDING THE VOLUMES OF SINGULAR WEAK LOG DEL PEZZO SURFACES

2013 ◽  
Vol 24 (13) ◽  
pp. 1350110 ◽  
Author(s):  
CHEN JIANG
Keyword(s):  
Upper Bound ◽  
General Position ◽  
Del Pezzo Surfaces ◽  
Minimal Resolution ◽  
Picard Number ◽  
Hirzebruch Surface ◽  
Del Pezzo Surface ◽  
Blowing Up ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

We give an optimal upper bound for the anti-canonical volume of an ϵ-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface. Furthermore, we consider blowing up several points on a Hirzebruch surface in general position and give some examples of smooth weak log del Pezzo surfaces.

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

2008 ◽  
Vol 19 (02) ◽  
pp. 173-191 ◽  
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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