Varieties of sums of powers Dedicated to the memory of

The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results (cf. [Sylvester 1851], [Hilbert 1888], [Dixon, Stuart 1906]) and some more recent results of Mukai (cf. [Mukai 1992]) are presented together with new results for the cases (n, d) = (3, 8), (4, 2), (5, 3). In the last case the variety of sums of 8 powers of a general cubic form is a Fano 5-fold of index 1 and degree 660. 0. Introduction 1. Apolarity and syzygies 2. Duality, projections and secants 3. Further examples 4. Cubic threefolds, first properties 5. Syzygies of 8 general points in P 6. Equations and geometry of the spinor varieties Sev and Sodd 7. The apolar Artinian Gorenstein ring of a general cubic threefold 8. Proof of the main results 9. Invariants of V SP (F, 8) References