By Julian Lowell Coolidge
A radical creation to the idea of algebraic airplane curves and their family to numerous fields of geometry and research. nearly totally limited to the homes of the final curve, and mainly employs algebraic approach. Geometric tools are a lot hired, even though, specifically these concerning the projective geometry of hyperspace. 1931 version. 17 illustrations.
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Extra resources for A treatise on algebraic plane curves
T0 ∂tn Recall that a hyperplane Ha = V ( ai ξi ) in the dual projective space (Pn )∨ is the point a = [a0 , . . , an ] ∈ Pn . The preimage of the hyperplane Ha under ∂f pX is the polar Pa (f ) = V ( ai ∂t ). i If X is nonsingular, the polar map is a regular map given by polynomials of degree d − 1. Since it is a composition of the Veronese map and a projection, it is a finite map of degree (d − 1)n . 1 Assume X is nonsingular. The ramification divisor of the polar map is equal to He(X). Proof Note that, for any finite map φ : X → Y of nonsingular varieties, the ramification divisor Ram(φ) is defined locally by the determinant of the linear map of locally free sheaves φ∗ (Ω1Y ) → Ω1X .
I−e0 −en |i|=d i0 in Ai x P 1 C .. C A . P i−en −en , i i A x n n i |i|=d where xi−es = 0 if i − es is not a non-negative vector. It is easy to interpret solutions of these equations as pairs (g, v) from above such that ∇(g)(v0 ) + He(f )(v0 ) · v = 0. 42) Since [v0 ] is a singular point of V (f ), ∇(f )([v0 ]) = 0. 10. This confirms that pairs (λf, µv0 ) are always solutions. 42). Its kernel consists of the pairs (λf, v) modulo pairs (λf, µv0 ). 42) give He(f )(v0 ) · v = 0. 43) We may assume that v0 = (1, 0, .
In the other two cases, the equation expresses that the second partials of the quartic are linearly dependent (see , pp. ) One can also consider the problem of a representation of several forms f1 , . . , fk ∈ S d (E ∨ ) as a sum of powers of the same set (up to proportionality) of linear forms l1 , . . , ls . This means that the forms share a common polar s-hedron. For example, a well-known result from linear algebra states that two general quadratic forms q1 , q2 in k variables can be simultaneously diagonalized.
A treatise on algebraic plane curves by Julian Lowell Coolidge