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It is a common wisdom that sometimes it helps to view ellipses, parabolas and hyperbolas in the real two-dimensional plane as actually `just the visible part' of a slightly large curve in the real projective plane (thought of as the ordinary plane with an extra projective line 'at infinitiy'). In particular, the same projective conic will look like an ellipse in some affine pieces of the projective plane, like a hyperbola in some others and like a parabola yet another small set of affine subsets. A nice unifying viewpoint. A popular way of expressing (part of) this is that parabola is just an ellipse but with one of its points on the line through infinity.

At the same time there is a different way of defining conics: as conflict sets.

A parabola is the set of points that are on equal distance from a point (the focus) and a line (the directrix) and an ellipse is the set of points at equal distance from a point (one of the two foci) and a circle (of which I don't know if it has a name).

Now looking at it very naively we could hope that these definitions also become the same when looking at it projectively, i.e. that every projective conic can be defined as the conflict line between a point and a different projective curve that (depending on at which affine piece we are looking right now) may look like either a line or a circle.

I have a feeling that in fact this is too naive. So here are two simpler questions that only revolve around the parabola-case:

  1. Every line and point not on it together define a parabola (as their conflict set). The parabola extends to a unique conic $C$ in the projective plane, the line extends to a unique projective line $l$ and the point $F$ is a perfectly fine point in the projective plane. Now suppose we are given the projective plane with $l$ and $F$ in it, but no info what affine piece was used for the original construction of the parabola from them. Can we still use $l$ and $F$ to define $C$ even if the notion of 'all points which have the same distance to $F$ and to $l$' does not seem to make too much sense anymore?

  2. Conversely every parabola in the affine plane uniquely determines its focus and directrix. So suppose that I start with a conic $C$ in the projective plane $P$. For every projective line $t$ tangent to $C$, the set $P \backslash t$ is an affine plane in which $C$ looks like a parabola. So in each of these affine pieces $C$ gives rise to a focus and a directrix. These directrices each extend to unique projective lines in $P$ again. Question: when I run over all infinitely many choices of $t$, how many different foci and how many different (projective) directrices do I obtain?

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  • $\begingroup$ Initially I was hesitant to attach the 'algebraic geometry' tag since it seems unlikely that the answer is best understood in terms of motivic cohomology, but after browsing the related questions in the side bar I felt that perhaps the tag might be acceptable for this kind of question on this site and would even help in attracting attention of the right people. $\endgroup$ – Vincent Jan 26 '18 at 21:29
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    $\begingroup$ Off the top of my head, you might be able to develop something via the pole-polar relationship of focus and directrix. $\endgroup$ – amd Jan 27 '18 at 0:40

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