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Hyperbolic Monopoles and Rational Normal Curves HYPERBOLIC MONOPOLES AND RATIONAL NORMAL CURVES Nigel Hitchin (Oxford) Edinburgh April 20th 2009 204 Research Notes A NOTE ON THE TANGENTS OF A TWISTED CUBIC B Y M. F. ATIYAH Communicated by J. A. TODD Received 8 May 1951 1. Consider a rational normal cubic C3. In the Klein representation of the lines of $3 by points of a quadric Q in Ss, the tangents of C3 are represented by the points of a rational normal quartic O4. It is the object of this note to examine some of the consequences of this correspondence, in terms of the geometry associated with the two curves. 2. C4 lies on a Veronese surface V, which represents the congruence of chords of O3(l). Also C4 determines a 4-space 2 meeting D. in Qx, say; and since the surface of tangents of O3 is a developable, consecutive tangents intersect, and therefore the tangents to C4 lie on Q, and so on £lv Hence Qx, containing the sextic surface of tangents to C4, must be the quadric threefold / associated with C4, i.e. the quadric determining the same polarity as C4 (2). We note also that the tangents to C4 correspond in #3 to the plane pencils with vertices on O3, and lying in the corresponding osculating planes. 3. We shall prove that the surface U, which is the locus of points of intersection of pairs of osculating planes of C4, is the projection of the Veronese surface V from L, the pole of 2, on to 2. Let P denote a point of C3, and t, n the tangent line and osculating plane at P, and let T, T, w denote the same for the corresponding point of C4. Further, let WXTB2 meet in Q, which is therefore a point of U, and let LQ meet Q. in R, R'. We show that R and R' represent the two lines PXP2 and (nv n2), and therefore that R, or R', is a point of V. The polar 3-space of Q with respect to I, being the same as that with respect to C4, meets C4 twice at each of Tx and T2, and therefore contains TX and T2. But this 3-space is the polar 3-space of LQ with respect to Q. Hence R and R', being conjugate to all the points of rx and T2, represent lines in S3 which intersect all lines of the pencils (Plt n^) (P2, TT2), and so must be. the lines PXP2, {nx, n2) as stated. 4. We now give a geometrical proof of the well-known result: a necessary and suffi- cient condition for four points on Ct to be equianharmonic is that the pole of the 3-space determined by them should lie on / (i.e. that the 3-space should touch /). Let A, B, G be three points on Ct, and let T be a collineation on CA, such that T(ABC) = (BCA). 3 Then T = 1, the identical collineation, and T(H1H2) = {HXH2), where H^, H2 are the Hessian pair ofA,B,C. Since C4 is a rational normal curve, there is a unique collinea- Z tion S of /S4 which induces T on C4, and 8 = 1. M F Atiyah, A note on the tangents to a twisted cubic, Proc. Camb. Phil. Soc. 48 (1952) 204–205 “.. The tangents at four points of a twisted cubic have a unique transversal if and only if the four points are equianharmonic”. RATIONAL NORMAL CURVES P1 Pn of degree n • ⊂ ... not contained in any hyperplane • = image by a projective transformation of z [1, z, z2, . , zn] • "→ Symmetric product Sn(P1) = Pn • Diagonal ∆ Sn(P1) = (x, x, . , x) : x P1 • ⊂ { ∈ } V = 2-dimensional vector space, SnV symmetric tensor prod- • uct Sn(P(V )) = P(SnV ), ∆ = [v v . v] : v V • { ⊗ ⊗ ⊗ ∈ } SymmetricSymmetric pproroductduct SSnn((PP11)) == PPnn •• DiagonalDiagonal ∆∆ SSnn((PP11)) == ((x,x,x,x,......,,xx)) :: xx PP11 •• ⊂⊂ {{ ∈∈ }} VV ==2-dimensional2-dim symplecticvectovector space,r space,SnV symmetricSnV symmetrictensortensoprod-r •• uctproduct SSnn((PP((VV)))) == PP((SSnnVV),), ∆∆ == [[vv vv ...... vv]] :: vv VV •• {{ ⊗⊗ ⊗⊗ ⊗⊗ ∈∈ }} rational normal curve C P(W ) defines an isomorphism • ⊂ W = H0(C, (n)=SnH0(C, (1)) = SnV ∗ ∼ O O ∗ SnV has a symplectic/ orthogonal (n odd/even) structure • EXAMPLES conic in P2 • twisted cubic in P3 • tangents to a twisted cubic Q4 P5 ... • ⊂ ⊂ ... lies in P4 Q4 • ∩ (S3V symplectic, P(S3V ) contact, twisted cubic Legendrian) • EXAMPLESEXAMPLES 2 conicconic inin PP2 •• 3 ttwistedwisted cubiccubic inin PP3 •• 4 5 tangentstangents toto aa ttwistedwisted cubiccubic QQ4 PP5 ...... •• ⊂⊂ ⊂⊂ 4 4 ...... lieslies inin PP4 QQ4 •• ∩∩ 3 3 ((SS3VV symplectic,symplectic, PP((SS3VV )) contact,contact, ttwistedwisted cubiccubic Legendrian)Legendrian) •• 204 Research Notes A NOTE ON THE TANGENTS OF A TWISTED CUBIC B Y M. F. ATIYAH Communicated by J. A. TODD Received 8 May 1951 1. Consider a rational normal cubic C3. In the Klein representation of the lines of $3 by points of a quadric Q in Ss, the tangents of C3 are represented by the points of a rational normal quartic O4. It is the object of this note to examine some of the consequences of this correspondence, in terms of the geometry associated with the two curves. 2. C4 lies on a Veronese surface V, which represents the congruence of chords of O3(l). Also C4 determines a 4-space 2 meeting D. in Qx, say; and since the surface of tangents of O3 is a developable, consecutive tangents intersect, and therefore the tangents to C4 lie on Q, and so on £lv Hence Qx, containing the sextic surface of tangents to C4, must be the quadric threefold / associated with C4, i.e. the quadric determining the same polarity as C4 (2). We note also that the tangents to C4 correspond in #3 to the plane pencils with vertices on O3, and lying in the corresponding osculating planes. 3. We shall prove that the surface U, which is the locus of points of intersection of pairs of osculating planes of C4, is the projection of the Veronese surface V from L, the pole of 2, on to 2. Let P denote a point of C3, and t, n the tangent line and osculating plane at P, and let T, T, w denote the same for the corresponding point of C4. Further, let WXTB2 meet in Q, which is therefore a point of U, and let LQ meet Q. in R, R'. We show that R and R' represent the two lines PXP2 and (nv n2), and therefore that R, or R', is a point of V. The polar 3-space of Q with respect to I, being the same as that with respect to C4, meets C4 twice at each of Tx and T2, and therefore contains TX and T2. But this 3-space is the polar 3-space of LQ with respect to Q. Hence R and R', being conjugate to all the points of rx and T2, represent lines in S3 which intersect all lines of the pencils (Plt n^) (P2, TT2), and so must be. the lines PXP2, {nx, n2) as stated. 4. We now give a geometrical proof of the well-known result: a necessary and suffi- cient condition for four points on Ct to be equianharmonic is that the pole of the 3-space determined by them should lie on / (i.e. that the 3-space should touch /). Let A, B, G be three points on Ct, and let T be a collineation on CA, such that T(ABC) = (BCA). 3 Then T = 1, the identical collineation, and T(H1H2) = {HXH2), where H^, H2 are the Hessian pair ofA,B,C. Since C4 is a rational normal curve, there is a unique collinea- Z tion S of /S4 which induces T on C4, and 8 = 1. 6246 24 R. RL. LE. ES. 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