scholarly journals On surfaces of class VII with numerically anticanonical divisor

2006 ◽  
Vol 128 (3) ◽  
pp. 639-670 ◽  
Author(s):  
Georges Dloussky
Keyword(s):  
Anticanonical Divisor

scholarly journals The family of lines on the Fano threefold V5

1989 ◽  
Vol 116 ◽  
pp. 111-122 ◽  
Author(s):  
Mikio Furushima ◽  
Noboru Nakayama
Keyword(s):  
Linear System ◽  
Betti Number ◽  
Invertible Sheaf ◽  
The Family ◽  
Anticanonical Divisor ◽  
Fano Threefold ◽  
Ample Invertible Sheaf

A smooth projective algebraic 3-fold V over the field C is called a Fano 3-fold if the anticanonical divisor — Kv is ample. The integer g = g(V) = ½(- Kv)3 is called the genus of the Fano 3-fold V. The maximal integer r ≧ 1 such that ϑ(— Kv)≃ ℋ r for some (ample) invertible sheaf ℋ ε Pic V is called the index of the Fano 3-fold V. Let V be a Fano 3-fold of the index r = 2 and the genus g = 21 which has the second Betti number b2(V) = 1. Then V can be embedded in P6 with degree 5, by the linear system |ℋ|, where ϑ(— Kv)≃ ℋ2 (see Iskovskih [5]). We denote this Fano 3-fold V by V5.

scholarly journals Tropical Lagrangians in toric del-Pezzo surfaces

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Jeffrey Hicks
Keyword(s):  
Lagrangian Submanifold ◽  
Del Pezzo Surfaces ◽  
Dimer Model ◽  
Mirror Pair ◽  
Wall Crossing ◽  
Lagrangian Tori ◽  
Anticanonical Divisor ◽  
Del Pezzo

AbstractWe look at how one can construct from the data of a dimer model a Lagrangian submanifold in $$(\mathbb {C}^*)^n$$ ( C ∗ ) n whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori $$L_{T^2}$$ L T 2 in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair $$(\mathbb {CP}^2{\setminus } E, W), {\check{X}}_{9111}$$ ( CP 2 \ E , W ) , X ˇ 9111 . We find a symplectomorphism of $$\mathbb {CP}^2{\setminus } E$$ CP 2 \ E interchanging $$L_{T^2}$$ L T 2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on $${\check{X}}_{9111}$$ X ˇ 9111 .

scholarly journals Rational Surfaces with Anticanonical Divisor not Reduced

2013 ◽  
Vol 21 (3) ◽  
pp. 229-240
Author(s):  
Jesús Adrian Cerda Rodriguez ◽  
Mustapha Lahyane ◽  
Osvaldo Osuna-Castro ◽  
Gioia Failla ◽  
Israel Moreno-Mejia
Keyword(s):  
Rational Surface ◽  
Effective Divisor ◽  
Canonical Divisor ◽  
Del Pezzo Surface ◽  
Multiplicity One ◽  
Irreducible Components ◽  
Complete Linear System ◽  
Anticanonical Divisor ◽  
Del Pezzo ◽  
Almost All

AbstractWe prove the finite generation of the monoid of effective divisor classes on a smooth projective rational surface X endowed with an anticanonical divisor such that all its irreducible components are of multiplicity one except one which has multiplicity two. In almost all cases, the self-intersection of a canonical divisor KX on X is strictly negative, hence - KX is neither ample nor numerically effective. In particular, X is not a Del Pezzo surface. Furthermore, it is shown that the first cohomology group of a numerically effective divisor vanishes; as a consequence, we determine the dimension of the complete linear system associated to any given divisor on X

scholarly journals Conic bundles that are not birational to numerical Calabi--Yau pairs

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
János Kollár
Keyword(s):  
Projective Plane ◽  
Number Field ◽  
Projective Variety ◽  
Conic Bundle ◽  
Anticanonical Divisor ◽  
Conic Bundles ◽  
Branch Curve

Let $X$ be a general conic bundle over the projective plane with branch curve of degree at least 19. We prove that there is no normal projective variety $Y$ that is birational to $X$ and such that some multiple of its anticanonical divisor is effective. We also give such examples for 2-dimensional conic bundles defined over a number field.

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