4 Constructions of Diagonal Quartic and Sextic Surfaces with Infinitely Many Rational Points (original) (raw)

Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

arXiv (Cornell University), 2014

In this note we construct several infinite families of diagonal quartic surfaces ax 4 + by 4 + cz 4 + dw 4 = 0, where a, b, c, d ∈ Z \ {0} with infinitely many rational points and satisfying the condition abcd =. In particular, we present an infinite family of diagonal quartic surfaces defined over Q with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type ax 6 + by 6 + cz 6 + dw i = 0, i = 2, 3, or 6, with infinitely many rational points.

Quartic surface, its bitangents and rational points

arXiv: Number Theory, 2020

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field K, the set of algebraic points in X(\overeline K) which are quadratic over a suitable finite extension K' of K is Zariski-dense.

On surfaces of general type with q = 5

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2012

We prove that a complex surface S with irregularity q(S) = 5 that has no irrational pencil of genus > 1 has geometric genus pg(S) ≥ 8. As a consequence, one is able to classify minimal surfaces S of general type with q(S) = 5 and pg(S) < 8. This result is a negative answer, for q = 5, to the question asked in [MP1] of the existence of surfaces of general type with irregularity q ≥ 4 that have no irrational pencil of genus > 1 and with the lowest possible geometric genus pg = 2q−3. This gives some evidence for the conjecture that the only irregular surface with no irrational pencil of genus > 1 and pg = 2q − 3 is the symmetric product of a genus three curve.

Rational points on certain elliptic surfaces

Acta Arithmetica, 2007

\ Q, and let us assume that deg f ≤ 4. In this paper we prove that if deg f ≤ 3, then there exists a rational base change t → ϕ(t) such that there is a non-torsion section on the surface E f •ϕ . A similar theorem is valid in case when deg f = 4 and there exists t 0 ∈ Q such that infinitely many rational points lie on the curve Et 0 : y 2 = x 3 + f (t 0 )x. In particular, we prove that if deg f = 4 and f is not an even polynomial, then there is a rational point on E f . Next, we consider a surface E g : y 2 = x 3 + g(t), where g ∈ Q[t] is a monic polynomial of degree six. We prove that if the polynomial g is not even, there is a rational base change t → ψ(t) such that on the surface E g•ψ there is a non-torsion section. Furthermore, if there exists t 0 ∈ Q such that on the curve E t 0 : y 2 = x 3 +g(t 0 ) there are infinitely many rational points, then the set of these t 0 is infinite. We also present some results concerning diophantine equation of the form x 2 − y 3 − g(z) = t, where t is a variable.

Rational surfaces in ℙ 4 containing a plane curve

Annali Di Matematica Pura Ed Applicata, 1997

The families of smooth rational surfaces in P4 have been classified in degree ? 10. All known rational surfaces in P4 can be represented as blow-ups of the plane P2. The fine classification of these surfaces consists of giving explicit open and closed conditions which determine the configurations of points corresponding to all surfaces in a given family. Using a

Density of rational points on diagonal quartic surfaces

Algebra & Number Theory, 2010

Let a, b, c, d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in P 3 defined by ax 4 + by 4 + cz 4 + dw 4 = 0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic topology.

Rational surfaces with many nodes

Compositio Mathematica, 2002

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n 5 b 2 À 2 disjoint smooth rational curves with self-intersection À2, where b 2 is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with p g ¼ 0 and K 2 ¼ 8; 9 which admit an automorphism of order 2.

A Class of Rational Surfaces in P

2003

In this paper, we obtain a complete classification of all rational surfaces embedded in P4 so that all their exceptional curves are lines. These surfaces are exactely the rational surfaces shown by I.Bauer to project isomorphicaly from P5 from one of their points, although no a priori reason is known why such a surface should be projectable in this way.

On the Brauer group of diagonal quartic surfaces

Journal of the London Mathematical Society, 2011

We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over Q to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of Q. The proof is based on the isomorphism of the Fermat quartic surface with a Kummer surface due to Masumi Mizukami.