## Keywords

Riemann Surface Tangent Plane Tangent Line Algebraic Curve Symmetric Pair
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.J. V. Poncelet,
*Traité des propriétés projectives des figures*, Mett-Paris 1822.Google Scholar - 2.C. G. J. Jacobi,
*Über die Anwendung der elliptischen Transcendenten auf ein bekanntes Problem der Elementargeometrie*, Gesammelte Werke, Vol.**I**(1881), pp. 278–293.Google Scholar - 3.P. Griffiths,
*Variations on a theorem of Abel*, Inventiones Math. Vol. 35 (1976), ppg. 321–390. This paper also contains a variant of Jacobi's discussion of the classical Poncelet problem.zbMATHCrossRefGoogle Scholar - 4.We shall allow the polygon to have self-intersections.Google Scholar
- 5.
- 6.We note that
*S*∩*T*_{p}(*S*) is a plane conic containing a line, and hence must be two lines through*P*as in Fig. 6. These lines are distinct since det*Q*#0.Google Scholar - 7.Since
*S*∩(*L*Ν*L'*) contains and therefore is equal to the plane conic*L+L'*, it follows that*L*Ν*L'*is the tangent plane to*S*at*L*∩*L'*. In particular,*L*_{0}, does not lie in*L*Ν*L'*nor pass through*L*∩*L'*.Google Scholar - 8.We will comment on the construction in ℝ
^{3}at the end of the paper.Google Scholar - 9.The vertices are where three or more planes meet. In Fig. 8 these are the vertices of the shaded quadrilaterals.Google Scholar
- 10.This is the formula
*K*_{D}=*K*_{M}*⊗[D]*_{D}for the canonical line bundle of a smooth divisor*D*on a complex manifold*M*.Google Scholar - 11.That the intersection is transverse follows from the equations for
*S*and*S'*given below.Google Scholar - 12.In this connection M. Berger pointed out two papers by Cayley giving explicit conditions for Poncelet polygon to be closed. The references are Philosophical Magazine, vol. VI (1853), 99–102. and Philosophical Trans. Royal Soc. London, vol. CLI (1861), 225–239.Google Scholar

## Copyright information

© Birkhäuser Verlag 1977