Varieties of sums of powers

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. [3 8], [14], [8]) and some more recent results of Mukai (cf. [24]) are present ed together with new results for the cases (n, d) = (3, 8), (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.