Review of the Elements of Intersection Theory For Two Dimensional Schemes: Exploring the Intersection Theory and its Applications in Algebraic Geometry and Topology

Abstract

We review here the elements of intersection theory fortwo dimensional schemes. In mathematics, intersection theory is a branch ofalgebraic geometry, where subvarieties are intersected on an algebraic variety,and of algebraic topology, where intersections are computed within thecohomology ring. The theory for varieties is older, with roots in Bézout'stheorem on curves and elimination theory. On the other hand the topologicaltheory more quickly reached a definitive form.For a connected oriented manifoldM of dimension 2n the intersection form is defined on the nth cohomology group(what is usually called the 'middle dimension') by the evaluation of the cupproduct on the fundamental class