Secants, Socles and Stability
Abstract
The projective space of symmetric tensors of degree d can be reinterpreted as a projective space of finite, graded Gorenstein rings with socle in degree d. Via a pair of explicit stability conditions (one for even values of d and one for odd values), the space of symmetric tensors is partitioned by Harder-Narasimhan filtration type. This is worked out explicitly for low degree examples in dimension three (the projective plane) and compared with the betti tables of the Gorenstein rings.
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