Enumerative geometry of planar conics

Abstract

In this thesis three problems from enumerative geometry will be solved. The first problem, Apollonius circles, which states how many circles are tangent to three given circles will be solved just using some polynomials and some algebra. It turns out that there are, with reasonable assumptions, eight of those. The second problem deals with how many lines intersect with four given lines in P^3 over C. In general, there are two of those. The last problem, how many conics are tangent to five given conics, is solved using blowups, the dual space, Buchberger's algorithm and the Chow group. It turns out that there are 3264 conics which satisfy this solution. Furthermore, it is easy to extend this solution to find how many conics are tangent to c conics, l lines and pass through p points where p + l + c = 5 and these points, lines and conics satisfy reasonable assumptions.