Zeta Functions of Conic Bundles and Del Pezzo Surfaces of Degree 4 over Finite Fields

2005 ◽  
Vol 5 (4) ◽  
pp. 919-926 ◽  
Author(s):  
S. Rybakov
Keyword(s):  
Finite Fields ◽  
Zeta Functions ◽  
Del Pezzo Surfaces ◽  
Conic Bundles ◽  
Del Pezzo

scholarly journals A Characterization of Some Fano 4-folds Through Conic Fibrations

2019 ◽  
Author(s):  
Pedro Montero ◽  
Eleonora Anna Romano
Keyword(s):  
Del Pezzo Surfaces ◽  
Conic Bundle ◽  
Conic Bundles ◽  
Del Pezzo ◽  
Bundle Structure

Abstract We find a characterization for Fano 4-folds $X$ with Lefschetz defect $\delta _{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $\rho _{X}-\rho _{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $\delta _{X}=3$.

scholarly journals Del Pezzo surfaces over finite fields and their Frobenius traces

2018 ◽  
Vol 167 (01) ◽  
pp. 35-60 ◽  
Author(s):  
BARINDER BANWAIT ◽  
FRANCESC FITÉ ◽  
DANIEL LOUGHRAN
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Del Pezzo Surfaces ◽  
Cubic Surface ◽  
Inverse Galois Problem ◽  
Complete Answer ◽  
Cubic Surfaces ◽  
Analogous Question ◽  
Special Cases ◽  
Del Pezzo

AbstractLet S be a smooth cubic surface over a finite field $\mathbb{F}$q. It is known that #S($\mathbb{F}$q) = 1 + aq + q2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.

scholarly journals Quartic del Pezzo surfaces over function fields of curves

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Brendan Hassett ◽  
Yuri Tschinkel
Keyword(s):  
Finite Fields ◽  
Function Fields ◽  
Projective Line ◽  
Total Space ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Numerical Invariants ◽  
Del Pezzo ◽  
Intermediate Jacobian

AbstractWe classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

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