Sketch of a verification of Schubert's number 5 819 539 783 680
of twisted cubics
Proceedings of the 1985 Rocca di Papa Conference on Space curves
Springer Lecture Notes in Mathematics 1266 (1987), 156--180.
From the MR review: "A rigorous derivation of the classical enumerative result of H. Schubert that there are 5 819 539 783 680 twisted cubic curves in Pk3 (k algebraically closed and of characteristic 0) tangent to 12 smooth quadric surfaces in general position. Moreover, the authors show that each cubic appears with multiplicity 1 in the count above. In characteristic p>0, Schubert's number represents a weighted number of cubics. (The weighting equals 1 if p does not divide 5 819 539 783 680.)"