On The Saturation Number For Cubic Surfaces And Hyprsurfaces

A fundamental theme in mathematics is the study of integral or rational points on algebraic varieties. For Fano varieties, the Manin conjecture predicts the distribution of rational points on the variety. Given a Fano variety X C Pn and a rational point x on the variety X, we may represent the point x as x= (x0,...,xn), where (x0,...,xn) ∈ Zn+1 and gcd(x0,...,xn)= 1. Define the height of the rational point x by H(x):=max{|xi|,0≤i≤n}. For a given Zariski open subset U C X, we may then defineThe Manin conjecture states that given X there exists an open subset U C X, such that for sufficiently large B, we havewhere cx> 0 is a positive constant,βx is a positive constant which equals 1 in the special case that X has dimension 2 and ρx denotes the rank of the Picard group of X.Meanwhile, Bourgain, Gamburd and Sarnak [4] developed the affine linear sieve to investigate the distribution of integral points with almost prime coordi-nates on varieties equipped with a group action. Subsequently, Nevo and Sarnak [55] and Liu and Sarnak [52] considered integral points with almost prime coor-dinates on certain homogeneous spaces and affine quadrics, respectively.Given a Fano variety X c Pn defined over Q, a rational point x on the variety X shall be called an almost prime point when the coordinates of the rational point have few prime factors. Then we may generalize the definition of the saturation number, which was introduced in [4], to investigate the distribution of almost prime points on the variety X. Define the saturation number r(X) to be the least number r such that the set of x ∈ X for which the product of the coordinates X0... xn has at most r prime factors, is Zariski dense in X.In this dissertation, we establish the finiteness of the saturation number for certain cubic surfaces and hypersurfaces.In Chapter 2 of this dissertation, we consider the saturation number for the Cayley cubic surface X1, which is a singular surface defined in P3 by the equation Combining the Hardy-Littlewood circle method and the theory of universal tor-sors, we prove that the saturation number for the Cayley cubic surface r(X1) satisfies 6≤r(X1)≤12. Similar technique applies to other singular cubic sur-faces. Let X2 ∈ P3 be the singular cubic surface given by the equation We show that the saturation number r(X2) satisfies r(X2)≤12.In Chapter 3, we are concerned with the Fermat cubic surface X3, which is defined in P3 by the equation Note that the Fermat cubic surface X3 is a smooth cubic surface. Applying Euler’s parametrisation of the solutions and the weighted sieve, we are able to show that r(X3)≤20.It is of interest to investigate whether we can establish a finite upper bound of the saturation number for a large family of smooth cubic surfaces.Sofos and the author considered the saturation number for smooth cubic surfaces containing two skew rational lines. In Chapter 4, we present our result about the saturation number for such surfaces. Suppose that X4 is a smooth cubic surface which contains two skew rational lines. Then we have the saturation number r(X3) satisfies r(X3)< 32. The main tools used include the conic bundle structure of the smooth cubic surface X4 and the weighted sieve.Moveover, we consider some particular cubic surface X5 containing two skew rational lines, which is defined by the equation For this surface, we are able to apply a result due to Green, Tao, and Ziegler instead of the weighted sieve to get a smaller saturation number. In fact, we establish that the saturation number r(X5) satisfies r(X5)≤10.In Chapter 5, we generalize the problem of the saturation number for cubic surfaces to cubic hypersufaces.As an illustration we have chosen the Fermat cubic threefold X6, which is a smooth cubic hypersurface defmed bySofos and the author apply the weighted sieve to suitable parametrisation of so-lutions to investigate the saturation number r(X6). We prove that the upper bound r(X6)≤42 is admissible.