Enumeration of genus-two curves with a fixed complex structure in ℙ² and ℙ³

The main concrete result of this paper is enumeration of genus-two curves with complex structure fixed in ℙ² and ℙ³. Along the way, rational curves with certain simple singularities are counted as well. While the methods described can be used to count positive-genus curves in some other cases, the most powerful direct applications of the machinery developed are to enumeration of rational curves with a very large class of singularities in projective spaces.