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Generalizing Pascal’s Mystic Hexagon Theorem Will Traves Department of Mathematics United States Naval Academy Mathematics and Statistics Colloquium Dalhousie University Halifax 18 APR 2013 Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 1 / 24 Ulterior Motives Really in Halifax to honor my father as he nears the end of his term as President at Dalhousie. Tom Traves Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 2 / 24 Start at the Beginning Line Arrangement due to Pappus of Alexandria (Synagogue; c. 340) Richter-Gebert: 9 proofs in Perspectives on Projective Geometry, 2011 Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 3 / 24 Pascal’s Mystic Hexagon Theorem Pascal: placed the 6 intersection points on a conic (1639) Converse: Braikenridge-Maclaurin Theorem Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 4 / 24 Why Mystic? Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 5 / 24 The Projective Plane Figure 16 The linking numbers of triangles 124 and 356 is 0 because they are not linked; 2 obtained by attaching a line at infinity P2 is a compactification of Rwithout apart from each other being caught like a chain link. Thicken line at infinity: P2 = disk ∪ Möbius band The linking numbers of triangles 246 and 135 is 1 because they are linked toge P2 can’t be embedded in R3 (Conway, Gordon, Sachs (1983): linked triangles in K6 ) If you add all of the linking numbers of all sets of triangles in this partic Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 6 / 24 Bézout’s Theorem Compactness of P2 allows us to count solutions: Theorem (Bézout) Any two curves, without common components, defined by the vanishing of polynomials of degrees d1 and d2 meet in d1 d2 points in P2 , suitably interpreted. Lines meeting an y=3 y=2 y=1 y=0 4x2+9y2=36 Traves (USNA) Generalizing Pascal’s Theorem Line m points Doubl tangen 4x2 + Halifax, 18 APR 2013 7 / 24 The 8 ⇒ 9 Theorem Theorem (Cayley-Bacharach-Chasles) If two cubics C1 and C2 meet in 9 distinct points, then any other cubic C through 8 of these points goes through the ninth too. Elementary proof on Terry Tao’s blog (July 15, 2011) Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 8 / 24 Proof of Pascal’s Theorem C1 : deg 3 curve consisting of red lines C2 : deg 3 curve consisting of blue lines C: deg 3 curve consisting of conic and line through 1 and 2 Since C goes through 8 points of C1 ∩ C2 , C goes through the ninth point too (by CBC Theorem). Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 9 / 24 From Pascal to Möbius Theorem (Möbius) Inscribe a polygon with 4k + 2 sides into a conic and consider the 2k + 1 points where opposite edges meet. If 2k of these points are collinear, then so is the last one. Inscribe two 2k -gons in a conic and associate edges cyclicly. The associated edges meet in 2k points and if 2k − 1 of these point are collinear, then so is the last one. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 10 / 24 Folklore Theorem Suppose that k red lines meet k blue lines in a set Γ of k 2 distinct points. If S = 0 is an irreducible curve of degree d through kd points of Γ then the remaining points lie on a unique curve C of degree t = k − d. Proof (Existence): R: deg k poly defining red lines. B: deg k poly defining blue lines. Pick P ∈ S \ Γ. Choose Fa,b = aR + bB to vanish at P. Then S = 0 is a component of Fa,b = 0 (by Bézout and S irred) and Fa,b /S defines C. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 11 / 24 d-Constructible Curves Definition A curve C of degree t is d-constructible if there exist k = d + t red lines and k blue lines so that: (1) red and blue lines meet in a set Γ of k 2 distinct points (2) dk points in Γ lie on a degree-d curve S (3) the remaining tk points of Γ lie on C. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 12 / 24 Density: definition Definition A curve C of degree t is d-constructible if there exist k = d + t red lines and k blue lines so that: (1) red and blue lines meet in a set Γ of k 2 distinct points (2) dk points in Γ lie on a degree-d curve S (3) the remaining tk points of Γ lie on C. Definition The d-construction is dense in degree t if almost every degree-t curve is d-constructible (i.e. there is a nonempty Zariski-open set U ⊂ P(C[x, y, z]t ) so that every degree-t curve in U is d-constructible). Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 13 / 24 First Example Example For any d, all lines are d-constructible. Arrange d + 1 points on the line and red and blue lines through these points (with distinct intersection points). Folklore Theorem ⇒ ∃ complementary curve of degree d. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 14 / 24 Second Example Example For any d, all irreducible conics are d-constructible. Arrange 2(d + 2) points, joined cyclicly by alternating red and blue lines, ensuring distinct intersection points. Folklore Theorem ⇒ ∃ complementary curve of degree d. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 15 / 24 Counting Dimensions Theorem (T-) If d ≥ 3, then the d-construction is not dense in degrees d + 4 or higher. The 2-construction is not dense in degrees five or higher. The 1-construction is not dense in degrees six or higher. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 16 / 24 Elliptic Curves Elliptic curves are genus 1 smooth curves (isomorphic to a real torus). The points on the curve form a group. Any elliptic curve can be embedded in P2 as a smooth cubic E with 0E a point of inflection at infinity. Group law: A + B + C = 0 ⇐⇒ A, B, C are collinear pts of E. Can use CBC Theorem to check that this group law is associative. The smooth cubics form a dense set of all cubics in P2 . Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 17 / 24 Elliptic Curves are d-constructible Theorem (T-) All elliptic curves are d-constructible so the d-construction is dense in degree 3. Need to arrange 3(d + 3) points on E with a blue and red line through each of them; construction has d + 4 parameters. Case d even: Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 18 / 24 Elliptic Curves are d-constructible: Case d odd Theorem (T-) All elliptic curves are d-constructible so the d-construction is dense in degree 3. Case d odd (c.f. Möbius): Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 19 / 24 Density Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 20 / 24 1-Constructible Curves Theorem (Roth and T-) The 1-construction is dense in degrees 4 and 5. deg 4 pf: Find f = (f1 , . . . , f15 ) : C17 → {1-constr deg-4 curves}. Thm of generic fiber: dim(imf ) + dim(gen fiber) = dim(domain). Suppose P ∈ fiber f −1 (c); for unit tangent vector v to fiber, ∂fi Dv f (P) = v = Jac(f )(P)v = 0. xj x=P Rank Jac(f )(P0 ) = 15 ⇒ dim(fiber) = 2 ⇒ dim(gen fiber) ≤ 2. Thus dim(im f) ≥ 15 = dim(deg 4 curves). Chevalley’s Thm ⇒ im(f) ⊃ dense open set of 1-constr deg-4 curves. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 21 / 24 Density Conjecture The d-construction is dense in degree 4 for all d. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 22 / 24 Strongly Inscribed Polygons Call a polygon P strongly inscribed in a curve C if every edge, extended to a line, meets C only in points p with exactly two edge lines passing through p. The d-construction is dense in degree t if and only if a dense set of degree-t curves admits a strongly inscribed polygon with 2(d + t) edges. Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 23 / 24 Strongly Inscribed Polygon Results Theorem (T-) Every elliptic curve admits a strongly inscribed polygon with 2k ≥ 8 edges. Easy to see that if degree C is odd then only polygons with an even number of edges can be strongly inscribed. Question Which degree 4 curves admit a strongly inscribed polygon with an odd number of sides? Traves (USNA) Generalizing Pascal’s Theorem Halifax, 18 APR 2013 24 / 24