Enumerative geometry

Remarks

Example: in $\mathbb{P}^3$, exactly $2$ lines meet four lines in general position. In the Grassmannian of lines, each condition is a hypersurface, and the product of four such classes is $2$ times the class of a point.

Example: there are $3264$ smooth conics in $\mathbb{P}^2$ tangent to five general conics. Steiner gave the answer $7776$, which counts degenerate double lines as well, and the correct number $3264$ is due to Chasles.

Non-example: the number of lines meeting four lines that all lie in one plane is infinite. Every line in that plane meets all four, so the conditions are not in general position.

Other classical numbers are $27$ lines on a smooth cubic surface and $2875$ lines on a general quintic threefold.