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.

Cubic surfaces over finite fields

2010 ◽  
Vol 149 (3) ◽  
pp. 385-388 ◽  
Author(s):  
PETER SWINNERTON–DYER
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Cubic Surface ◽  
Cubic Surfaces

Let V be a nonsingular cubic surface defined over the finite field Fq. It is well known that the number of points on V satisfies #V(Fq) = q2 + nq + 1 where −2 ≤ n ≤ 7 and that n = 6 is impossible; see for example [1], Table 1. Serre has asked if these bounds are best possible for each q. In this paper I shall show that this is so, with three exceptions:

scholarly journals Automorphisms of cubic surfaces in positive characteristic

2019 ◽  
Vol 83 (3) ◽  
pp. 15-92
Author(s):  
Игорь Владимирович Долгачев ◽  
Igor Vladimirovich Dolgachev ◽  
Alexander Duncan ◽  
Alexander Duncan
Keyword(s):  
Isomorphism Class ◽  
Normal Forms ◽  
Positive Characteristic ◽  
Automorphism Groups ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Intermediate Step ◽  
Cubic Surface ◽  
Cubic Surfaces ◽  
Del Pezzo

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero. Bibliography: 29 titles.

scholarly journals Cohomology of the universal smooth cubic surface

2020 ◽  
Author(s):  
Ronno Das
Keyword(s):  
Finite Field ◽  
Cubic Surface ◽  
Cubic Surfaces ◽  
Universal Family ◽  
Rational Cohomology

Abstract We compute the rational cohomology of the universal family of smooth cubic surfaces using Vassiliev’s method of simplicial resolution. Modulo embedding, the universal family has cohomology isomorphic to that of $\mathbb{P}^2$. A consequence of our theorem is that over the finite field $\mathbb{F}_q$, away from finitely many characteristics, the average number of points on a smooth cubic surface is q2+q + 1.

scholarly journals Plane quartics with a Galois-invariant Steiner hexad

2019 ◽  
Vol 15 (05) ◽  
pp. 1075-1109 ◽  
Author(s):  
Andreas-Stephan Elsenhans ◽  
Jörg Jahnel
Keyword(s):  
Del Pezzo Surfaces ◽  
Inverse Galois Problem ◽  
Plane Quartics ◽  
Del Pezzo

We describe a construction of plane quartics with prescribed Galois operation on the 28 bitangents, in the particular case of a Galois-invariant Steiner hexad. As an application, we solve the inverse Galois problem for degree two del Pezzo surfaces in the corresponding particular case.

Dickson Polynomials Over Finite Fields and Complete Mappings

1987 ◽  
Vol 30 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Gary L. Mullen ◽  
Harald Niederreiter
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Permutation Polynomials ◽  
Dickson Polynomials ◽  
Complete Mapping ◽  
Dickson Polynomial ◽  
Special Cases ◽  
Polynomials Over Finite Fields ◽  
Complete Mappings ◽  
Special Case

AbstractDickson polynomials over finite fields are familiar examples of permutation polynomials, i.e. of polynomials for which the corresponding polynomial mapping is a permutation of the finite field. We prove that a Dickson polynomial can be a complete mapping polynomial only in some special cases. Complete mapping polynomials are of interest in combinatorics and are defined as polynomials f(x) over a finite field for which both f(x) and f(x) + x are permutation polynomials. Our result also verifies a special case of a conjecture of Chowla and Zassenhaus on permutation polynomials.

scholarly journals The Effect of Points Fattening on del Pezzo Surfaces

2020 ◽  
Author(s):  
Magdalena Lampa-Baczyńska
Keyword(s):  
Complex Numbers ◽  
Del Pezzo Surfaces ◽  
Minimal Growth ◽  
Complete Answer ◽  
Initial Sequence ◽  
Blowing Up ◽  
Del Pezzo

Abstract In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces $$\mathbb {S}_r$$ S r arising by blowing-up of $$\mathbb {P}^2$$ P 2 at r general points, with $$ r \in \{1, \dots , 8 \}$$ r ∈ { 1 , ⋯ , 8 } . Basic questions when studying the problem of points fattening on an arbitrary variety are what is the minimal growth of the initial sequence and how are the sets on which this minimal growth happens characterized geometrically. We provide a complete answer for del Pezzo surfaces.

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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