Poncelet Porisms and Beyond: Integrable Billiards, by Vladimir Dragović

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By Vladimir Dragović

The aim of the ebook is to offer, in a whole and accomplished means, components of present study interlacing round the Poncelet porism: dynamics of integrable billiards, algebraic geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics. an important effects and concepts, classical in addition to glossy, hooked up to the Poncelet theorem are provided, including a historic evaluation reading the classical principles and their ordinary generalizations. detailed recognition is paid to the conclusion of the Griffiths and Harris programme approximately Poncelet-type difficulties and addition theorems. This programme, formulated 3 a long time in the past, is aimed to realizing the higher-dimensional analogues of Poncelet difficulties and the conclusion of the substitute process of upper genus addition theorems.

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On the other hand, to a curve in C2 that is given by polynomial p(x, y), we can naturally join the following curve in CP2 : P (x, y, z) = z d p x y , , z z d = deg p(x, y). Thus, projective and affine representations of an irreducible curve are completely equivalent and it is easy to switch between them. 35. Let P (x, y, z) be a homogeneous polynomial and P (x0 , y0 , z0 ) = 0, z0 = 0. If p(x, y) = P (x, y, 1), prove that ∂P ∂P ∂P (x0 , y0 , z0 ) = (x0 , y0 , z0 ) = (x0 , y0 , z0 ) = 0 ∂x ∂y ∂z is equivalent to ∂p ∂x x0 y 0 , z0 z0 = ∂p ∂y x0 y0 , z0 z0 = 0.

Pk (x, y) is a product of k linear factors, each one determining a tangent line to the curve at p. If all these tangents are distinct, then we say that p is an ordinary k-tuple point. 31. Determine all singular points of the curves listed at the beginning of this section and find the tangents at these points. Algebraic curves in the complex projective plane An algebraic curve of degree d and a line in C2 , generally, have d intersecting points, counting multiplicities. Consider a point P = (x0 , y0 ) ∈ C.

If p is a regular point, then the tangent to the curve at this point is given by the equation p1 (x, y) = 0. Otherwise, let k > 1 be the smallest number such that pk (x, y) ≡ 0. Then we will say that P is a k-tuple point on C. pk (x, y) is a product of k linear factors, each one determining a tangent line to the curve at p. If all these tangents are distinct, then we say that p is an ordinary k-tuple point. 31. Determine all singular points of the curves listed at the beginning of this section and find the tangents at these points.

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