Download Topics in Classical Algebraic Geometry - I by Igor V. Dolgachev PDF
By Igor V. Dolgachev
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The theory of apolarity is one of the forgotten topics of classical algebraic geometry. It originates from the works of Rosanes [144] and Reye [141]. We refer for survey of classical results to [133] and to a modern exposition of some of these results to [54] which we followed in these notes. 1 Self-polar triangles The Veronese quartic surface Recall that the Veronese variety is defined to be the image of the map P(E ∗ ) → P(S d E ∗ ), V (L) → V (Ld ), where L is a nonzero linear form on E. 1) where v is a nonzero vector in E.
Tn ). Let A be the matrix of size (n + 1) × d whose ith column is formed by the coefficients of li (defined, of course up to proportionality). Let ∆I be the maximal minor of A corresponding to a subset I of [1, . . , d] and fI be the product of linear forms li , i ∈ I. Show that X 2 2 He(f ) = (−1)n (d − 1)f n−1 ∆I fI . I 39 EXERCISES ([126], p. 660). 7 Let n = 2. Assume He(V (f )) = P2 . Show that f is the union of concurrent lines. 8 Show that the locus of the points on the plane where the first polars of a plane curve X are tangent to each other is the Hessian of X and the set of common tangents is the Cayleyan curve .
It is a reflexive relation on the set of lines. Obvioulsy, two triangles are conjugate if and only if each of the sides of the first triangle is conjugate to a side of the second triangle. 2. PONCELET RELATION Now let us consider the following problem. Given two triangles without common sides, find a conic C such that the triangles are conjugate to each other with respect to the conic C. Assume that the first triangle is formed by the coordinate lines ti = 0. 4) it is easy to get a necessary and sufficient condition for this to be true.