My favorite mind-bending phenomenon to mention to future high school teachers is the so-called Euler-Cramer paradox. The excellent and readable textbook by C. G. Gibson, Elementary Geometry of Curves, Cambridge University Press, 1998, does briefly discuss this (as do some more advanced textbooks such as Hartshorne), but not by name, and I know of no algebraic geometry textbooks which take the time to give what I feel would be appropriate emphasis to it.

This "paradox" is IMO so interesting that I'll take the trouble to try to explain it.
In the first pass I'll deliberately omit something--- see if you can spot it!

Stirling proved in 1717 that a set of $n(n+3)/2$ points in $CP^2$ (the complex projective plane) determine a unique degree n curve, namely the unique degree n curve passing through all $n(n+3)/2$ points. (Stirling's theorem is not hard to "prove" by considering the equation of the curve.)

So a set of 9 points determines a unique cubic curve. In other words, loosely speaking, the information needed to specify a cubic curve is at most the information needed to specify 9 points.

McLaurin proved in 1720 that a degree m curve intersects a degree n curve in $m \, n$ points. This is known as Bezout's theorem even though Bezout's proof was neither first nor correct! (This theorem is not hard to "prove" either.)

So a pair of cubic curves intersects in 9 points. In other words, loosely speaking, the information needed to specify 9 points is less than the information needed to specify a cubic curve.

McLaurin noticed that these results contradict one another! The first says 9 points determine a unique cubic curve, while the second says a pair of distinct cubics determines nine points--- yet only one cubic is supposed to pass through these nine.

Similarly, 14 points determine a unique quartic curve, but a pair of distinct quartic curves intersects in 16 points. Even worse! Is it $I \leq 14$ or $I > 16$?

Around 1750, Cramer and Euler independently noticed the same problem, and were apparently the first to suggest that the problem is cured by adding the phrase "points in general position" to the statement of Stirling's theorem and adding the phrase "a generic pair of distinct curves" to the statement of the McLaurin-Bezout theorem.

Here, "general position" is a famously slippery phrase, but the idea is that a "generic" set of points won't satisfy any algebraic conditions which might mess up the desired conclusion, so you should define "GP" to mean whatever you need it to mean in order to eliminate special cases!

Plucker proved in 1828 that if you delete any one point from a set of $n \, (n+3)/2$ points in GP, then an entire pencil of degree n curves passes through the remaining points, and any pair of these will intersect in $n^2$ points. In other words, the remaining points determine $(n-1)(n-2)/2$ additional points, all lying on the original cubic curve, which form a non-generic set of $n^2$ "pinch points" which common to a pencil of degree n curves.

So: a pair of cubic curves intersect in 9 points, but these are nongeneric. A generic set of 9 points determines a unique cubic, but omitting any one of these allows us to pass a one-parameter family (pencil) of cubics through the remaining 8. These then determine a unique additional point, forming a nongeneric set of 9 pinch points common to the entire pencil; that is, any pair of curves from the pencil intersect in precisely these 9.

Similarly, a pair of quartic curves intersect in 16 points, but these are nongeneric. A generic set of 14 points determines a unique quartic, but omitting any one allows us to pass a one-parameter family (pencil) of quartics through the remaining 13 points. These 13 points determine uniquely 3 additional points, forming a non-generic set of 16 pinch points common to the entire pencil; that is, any pair of curves from the pencil intersect in precisely these 16. These 16 pinch points are in fact multiply-special, in the sense that no subset of 14 is in GP!

(Actually, even these two statements might overlook some additional "genericity" requirements--- this stuff can get rather tricky!)

Turning things around, we can say that the $n^2$ points determined in the McLaurin-Bezout theorem satisfy a special relationship, which we might call the Cramer-Euler-McLaurin-Plucker (CEMP) property. Too bad Cramer was not I, eh? .oO

This kind of reasoning lead by the end of the nineteenth century to the development by Hilbert, Noether, and others, of what we now call homological algebra, which provides powerful algebraic tools capable of expressing various senses in which a finite set of points can be non-generic.

Take (almost) any cubic curve, choose (almost) any nine points on this curve, and delete any one of these nine. The remaining eight determines a unique CEMP nonet lying on the original curve (but "almost certainly" not the same as the original nonet). So CEMP nonets abound, and similarly for higher degrees.

Suppose we have a CEMP $n^2$-tuple. If we apply a projective transformation, the result will be a new CEMP $n^2$-tuple. So this relation is a projective invariant.

According to John Baez (keep your eye out for forthcoming Weeks), the key to understanding Kleinian geometry is to consider q-ary relations which are invariant under some group action--- in this case, the projective group $PGL(3,C)$ in its natural action on $CP^2$. I have often decried the effort expended these days on quantizing gravity, compared to that spent on explicating probability. I have suggested seeking an algebraic theory in which probability emerges from the notion of "generic" (think "probability one", and compare the notion of "spin networks").

Clearly, there's much more to arrangements of points in the plane than most people realize!

Maybe others here also have candidates for fun topics they'd like to see better known among high school math teachers?

(Note: I originally posted this at the end of another thread, but I have moved it to a new thread because I don't want it to be buried.)

Last edited:

matt grime
Homework Helper
I'm waiting for the next installment, which _has_ to mention the Hilbert Scheme.

mathwonk
Homework Helper
it seems odd the same "paradox" was not noticed in the case of planes in space, since the assertion that 3 points determine a plane similarly "contradicts" the statement that two planes meet in a line, containing many triples of points.

this is a good illustration of the concepts of linear algebra, for students wanting motivation, since the hyperplanes in the space of cubics, defined by vanishing at points of the plane, of course need not always be independent.

the map taking a point to the hyperplane of cubics vanishing at that point, defines a map from P^2 to P^9, and the iobservation is that not all finite sets of points on that image surface are independent.

there are some cases where this does not occur i think? maybe for a twisted cubic in 3 space, all points of the curve are independent? well lets translate it into a similar statement. we would be looking at the vanishing of uh, cubic? polynomials on the line?

and since no two cubic polynomials can vanish at the same three points, i guess indeed there are collinear triples on this space curve? maybe.

Hilbert's take on non-generic sets of points

I'm waiting for the next installment, which _has_ to mention the Hilbert Scheme.
You guessed where I am heading, but an intermediate stop would be Schenck, Computational Algebraic Geometry (especially chapter 7).

it seems odd the same "paradox" was not noticed in the case of planes in space, since the assertion that 3 points determine a plane similarly "contradicts" the statement that two planes meet in a line, containing many triples of points.
Good point, I should have mentioned this first.

Readers knowledgeable about Schubert calculus will know that that linear subspaces sitting inside higher degree varieties plays an important role in studying spaces of curves of a given degree. And in a sense Schubert calculus concerns computations of a cohomology ring (or more generally of a Chow ring).

Last edited:
mathwonk
Homework Helper
i wonder if the map from P^2 to P^9 is the one given by cubic polynomials, i.e. to the degree 9 del pezzo surface in P^9. I suppose so, i.e. what else!

so the question becomes whether all sets of points of cardinality k on a degree 9 del pezzo surface in P^9 are independent, for various values of k < 10.

mathwonk
Homework Helper
by the way, the extra points determined by a choice of generic ones, determine the genus of the curve. so a cubic has genus one and a quartic has genus 3!

use this method to determine the genus of a plane quintic.

Computing genus from degree of complex projective plane curve

by the way, the extra points determined by a choice of generic ones, determine the genus of the curve. so a cubic has genus one and a quartic has genus 3!
Good point, I should have mentioned that. See Frances Kirwan, Complex Algebraic Curves, section 4.3 for a proof of the formula $g=(n-1) \, (n-2)/2$.
use this method to determine the genus of a plane quintic.
Plugging in, a cubic, quartic, quintic complex projective plane curve, treated as a Riemann surface, is homeomorphic a sphere with 1,3,6 handles, respectively (i.e., homeomorphic to the surface of a one-holed, three-holed, or six-holed pretzel).

Mathwonk, do you know a proof directly related to Plucker's theorem?

Last edited:
the map taking a point to the hyperplane of cubics vanishing at that point, defines a map from P^2 to P^9, and the iobservation is that not all finite sets of points on that image surface are independent.
For benefit of lurkers, you (we?) should probably slow down a bit and explain Veronese, Segre, rational normal scrolls... But I don't want to steal your thunder if you are planning to explain this.

Edit: this whole thread should really be moved in "Linear and Abstract Algebra" since it hasn't developed in the manner expected. See https://www.physicsforums.com/showthread.php?p=1413810#post1413810 for some background to homological algebra, a repost of a classic three-part sci.math post by james dolan in which jd explains that anyone who has ever carried a digit was using cohomology and didn't even know it

Last edited:
My favorite mind-bending phenomenon to mention to future high school teachers is the so-called Euler-Cramer paradox.

Maybe others here also have candidates for fun topics they'd like to see better known among high school math teachers?
Are you suggesting a high school math teacher ought to understand this paradox, or just that this paradox ought to be mentioned to high school math teachers? What advantage or positive at all would there be in a high school math teacher knowing this (which of course is different from it just being mentioned)? I'm also curious, how much time have you spent teaching high school math and what kind of school(s) were they?

I'd be happy if high school math teachers just knew the basics. I've seen some divide an inequality by a negative, or multiply by a variable which might have been negative, without worrying about the sign. Just real basic errors or teaching from the book and needing the answers in the teacher's edition because their degree is in physical education.

Thanks for posting that. A nice intro to the type of problems algebraic geometry is concerned with.

Bob Clark

I didnt understand the last para of your post completely, but from what I gather, n(n+3)/2 points are required to completely define a n-degree curve, and if you remove one point from the n(n+3)/2 points, you get a family of n-degree distinct curves which differ due to the selection of the final point. Finally, if you have a m-degree and n-degree curve, they will intersect in mn points.

Generalizing this to two n-degree curves, where m=n, you get $$n^2$$ points of intersection. And the paradox is, that two curves can intersect in 9 points for two 3-degree curves but 9 is the minimum amount of points required to define a 3-degree curve, so something seems to be shoddy. But those nine points are not the same for the two curves. I mean, that those nine points of intersection uniquely define a third 3-degree unique curve. Where's the contradiction?

And I guess you didnt choose 2-degree curves because that particular problem doesnt arrive there. But even so, where's the contradiction?

Recapitulation

The "paradox" is only apparent. I deliberately stated the first two theorems with a crucial omission in order that the reader might briefly share McLaurin's puzzlement. To wit:

Stirling proved in 1717 that a set of $n(n+3)/2$ points in $CP^2$ (the complex projective plane) determine a unique degree n curve, namely the unique degree n curve passing through all $n(n+3)/2$ points.
...
McLaurin proved in 1720 that a degree m curve intersects a degree n curve in $m \, n$ points.
The McLaurin-Euler-Cramer "paradox" is this: the McLaurin-Bezout theorem says that two distinct cubic plane curves intersect in 9 points, but Stirling's theorem says that 9 points should determine a unique cubic curve; that is, a unique cubic should pass through these 9 points, yet we have two cubics (actually an entire one-parameter family of cubics) passing through these points. Similarly, the McLaurin-Bezout theorem says that two distinct quartic plane curves intersect in 16 points, but Stirling's theorem says that any 14 of these 16 points should determine a unique quartic curve; that is, a unique quartic should pass through any 14 of our 16 points, yet we have two quartics (actually an entire one-parameter family of quartics) passing through all 16 points. Similarly for higher degrees.

As Euler and Cramer independently observed, the "paradox" is resolved when we notice that Stirling theorem's tacitly assumes that the $n(n+3)/2$ points are "in general position" wrt each other, whereas the $n^2$ points obtained by intersecting two degree n curves are manifestly not "in general position"---if you like, this is precisely what McLaurin's observation really shows.

Plucker proved in 1828 that if you delete any one point from a set of $n \, (n+3)/2$ points in GP, then an entire pencil of degree n curves passes through the remaining points, and any pair of these will intersect in $n^2$ points. In other words, the remaining points determine $(n-1)(n-2)/2$ additional points, all lying on the original cubic curve, which form a non-generic set of $n^2$ "pinch points" which common to a pencil of degree n curves.
This is interesting on so many levels! First, it implies the existence of a whole sequence of rather subtle r-ary relations, "in general position", for arbitrarily large r, which are invariant under the symmetry group of the projective plane (see John Baez's recent "weeks" for why invariant relations are so important.) Second: consider the notion of "choosing points at random" in the plane or on a degree n curve (a geometrically natural notion which was well known to Euler). Plucker's result shows that the question of whether or not an r-tuple of points stand "in general position" wrt each other is actually rather subtle. Now think about this: how is the notion of "choosing r points at random" related to the notion of "r points standing in general position wrt each other"? Third: as mathwonk pointed out (see post #6), $(n-1)(n-2)/2$ happens to be the genus of our curve. Fourth: when Euler and Cramer were active, the notion of higher dimensional geometry was still regarded as philosphically suspect, even though the space of degree n-curves used by McLaurin is merely a higher dimensional projective space, and as mathwonk's remark (post #3, first paragraph) shows, this point of view sheds further light on the McLaurin-Euler-Cramer phenomenon. Some might say that the real paradox is that geometers took so long to regard n-dimensional projective spaces as having no less mathematical validity than the projective plane. Fifth: Euler knew all of the math required to invent classical information theory, eighteenth century geometers routinely appealed to the notion of "degrees of freedom" which is essentially information theoretic, and Euler had studied geometric probability problems. So I suggest that the real paradox is why Euler's study of the McLaurin "paradox" did not lead him to invent the information theory which was eventually discovered by Shannon after the passage of almost a further two centuries. How different history would have been had not one of Euler's grandchildren distracted him at the crucial moment! (See the classic book of biographical studies by my formally non-existent former colleague, E. T. Bell, Men of Mathematics.)

And sixth: as these musings show, the study of algebraic plane curves, a topic still routinely taught as part of "high school geometry" in most developed countries, conceals many subtleties and as it were is trying to usher us into the world of higher dimensional projective geometries, invariant relations of higher arity, algebraic topology, homological algebra (see matt grime's comment about Hilbert schemes in post #2), information theories, and so on. I think that's rather cool!

Last edited:
Whoa. Im sorry. This is way way beyond me. Ive just started college. Maybe in another four years I might have some clue of what youre talking about, but you totally lost me in your 4th and 5th para. What does "in general position" mean?