what are some applications of modern algebraic geometry to conic sections?

Question

The simplest non-trivial example of an algebraic curve is probably a conic section (ellipse, parabola and hyperbola). At the same time, we also know that development in advanced theory can provide new insights even to the most elementary (non-trivial) examples, e.g. Milnor's Topology from the Differentiable Viewpoint contains a proof of the fundamental theorem of algebra. Brouwer's fixed point theorem has an elementary statement, but it's far more than the scope of high school-levelled discussion of disks and balls etc.. So now this question naturally arises: what are some applications of the modern theory of algebraic geometry (Grothendieck, Serre, Mumford etc..) to the theory of conic sections, or more generally, to elementary mathematics?

Answer 1

A) If you take modern in a comprehensive sense, algebraic geometry has certainly thrown a new light on such classical topics as:
1) Pascal's theorem on the mystic hexagram (Fulton page 62).
2) Poncelet's porism on the closing of polygons inscribed in a conic: here is a sophisticated modern proof.
3) Steiner's problem ( the rigorous determination of the number, namely 3264, of conics tangent to five given conics) can be solved by using modern intersection theory . Here is a survey (in French, alas) .

B) But there are also really new results, with no classical counterpart, using the arsenal of modern algebraic geometry : for example Hartshorne's ellipse.