milan - Milan j. math. 72 (2004), 163187 DOI...

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Milan j. math. 72 (2004), 163–187 DOI 10.1007/s00032-004-0029-2 c ± 2004 Birkh¨auser Verlag Basel/Switzerland Milan Journal of Mathematics Dual Homogeneous Forms and Varieties of Power Sums Igor V. Dolgachev Abstract. We review the classical definition of the dual homogeneous form of arbitrary even degree which generalizes the well-known notion of the dual quadratic form. Following the ideas of S. Mukai we apply this construction to the study of the varieties parametrizing representations of a homogeneous polynomial as a sum of powers of linear forms. Mathematics Subject Classification (2000). Primary 14N15; Secondary 14J45, 15A72. Keywords. Apolarity, homogeneous forms, sums of powers, Fano three- folds. 1. Introduction A well-known theorem from linear algebra asserts that a nondegenerate quadratic form F 2 on a complex vector space V of dimension n +1canbe written as a sum of n + 1 squares of linear forms l i . The linear forms l i ’s considered as vectors in the dual space V are mutually orthogonal with respect to the dual quadratic form ˇ F 2 on the space V . For more than hun- dred years it has been a popular problem for algebraists and geometers to search for a generalization of this construction to homogeneous forms F d on V of arbitrary degree. It is known as the Waring problem or the canonical forms problem for homogeneous forms. The main object of the study is the variety of sums of powers VSP( F d ; N ) o parametrizing all representations of F d as a sum of powers of N linear forms. According to the traditional Research supported in part by NSF Grant DMS 0245203.
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164 Igor V. Dolgachev Vol. 72 (2004) definition, VSP( F d ; N ) o is the subset of the projective space P S N V of ho- mogeneous polynomials of degree N on V equal to the product of linear forms l 1 · ...l N such that F d = l d i . The zero sets V ( l i )o ftheforms l i ’s are hyperplanes in P V ,andtheset V ( l 1 ) ,...,V ( l N ) was classically known as the polar polyhedron of F d . Its faces V ( l i ) can be viewed as points in the dual space P V ,andthevar ietyVSP( F d ; N ) o is the subvariety of the symmetric power P V ( N ) of P V parametrizing the polar polyhedra of F d . The varietes VSP( F d ; N ) o were intensively studied in the classical algebraic geometry and the invariant theory in the works of A. Dixon, F. Palatini, T. Reye, H. Richmond, J. Rosanes, G. Scorza, A. Terracini, and others. How- ever, the lack of techniques of higher dimensional algebraic geometry did not allow them to give any explicit construction of the varieties VSP( F d ; N ) o or to study a possible compactification VSP( F d ; N )ofVSP( F d ; N ) o (except in the case n = 1 and a few cases where VSP( F 2 ,N ) o is a finite set of points). The first explicit construction of VSP( F d ; N ) o was given by S. Mukai in the cases ( n, d, N )=(2 , 2 , 3) , (2 , 4 , 6) , (2 , 6 , 10) for a general polynomial F d .
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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milan - Milan j. math. 72 (2004), 163187 DOI...

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