Convex Hull Generated By Chern Numbers Of Complete Intersection Surfaces Of General Type

Let X be a smooth projective surface of general type. Let c12 and χ denote the first Chern number and Euler characteristic of X respectively. The geography problem for surfaces of general type is to determine all possible values of (c12, χ)-This has a history of research in algebraic geometry. The famous Bogomolov-Miyaoka-Yau inequality states thatIn [Per], Ulf Persson proved that c12≤ 8χ for smooth complete intersection surfaces (SCI surfaces).In this thesis, we refine Persson result by determining the convex hull of all points (c12, χ) of SCI surfaces. Let X be a SCI surface in Pn+2. We first determine for each fixed n, the convex hull Σn of (c12,χ). We prove that Σn is unbounded and for n=1,2,3,4,5 its upper boundary consists of infinite line segments whose slopes form a convergent sequence while the lower boundary is simply given by a ray. For n≥ 6, Σn is bounded by two rays. We then deduce the convex hull of (c12,χ) for all SCI surfaces. For each n, we describe all linear inequalities satisfied by c12 and X for SCI surfaces in Pn+2. We also give a direct proof of those inequalities. Linear inequalities for Chern numbers are established for all SCI surfaces.