These varieties are studied via apolarity and syzygies.
Varieties of sums of powers
The apolar Artinian Gorenstein ring of a general cubic threefold 8.
Varieties of sums of powers
More generally we say that homogeneous forms f ∈ S and D ∈ T are apolar if f (D) = D(f ) = 0 (According to [Salmon 1885] the term was coined by Reye).
Varieties of sums of powers
In particular the socle of AF is 1-dimensional, and AF is indeed Gorenstein and is called the apolar Artinian Gorenstein ring of F ⊂ Pn .
Varieties of sums of powers
We call a subscheme Γ ⊂ ˇPn apolar to F , if the homogeneous ideal IΓ ⊂ F ⊥ ⊂ T .
Varieties of sums of powers
Similarly, of course, for a general form of even degree d = 2k , the apolar Artinian Gorenstein ring is a complete intersection AF ∼= C[∂0 , ∂1 ]/(a, b) with dega = degb = k + 1.
Varieties of sums of powers
Thus Γ is apolar to FD = V (D(f )), i. e. to all multiple partials of f .
Varieties of sums of powers
F e (Γ) is contained in a ﬁber of πF Therefore, when F ⊥ e over a linear space of e ( ˇPn ) this e is an embedding the apolar set Γ determine a s−secant PdΓ (e) to πF dimension dΓ (e).
Varieties of sums of powers
So as soon as e ≤ de (s) In fact, the image πF e ( ˇPn ) of dimension any scheme of length s on the line will contribute to the variety of s-secant spaces to πF de (s), while such schemes are not apolar to F .
Varieties of sums of powers
The general criterion for a subscheme Z to be apolar to F can be weakened to a useful suﬃcient criterium: Consider a scheme Z ⊂ ˇPn of length s such that the span of πF e (Z ) has dimension de (s).
Varieties of sums of powers
IZ is generated by forms of degree e, then IZ ⊂ F ⊥ and Z is apolar to F .
Varieties of sums of powers
For a general cubic, we then started to look for the variety of apolar elliptic quintic curves.
Varieties of sums of powers
With MACAULAY we computed examples of apolar Artinian Gorenstein rings for cubics f with general syzygies, and found a rank 10 quadratic relation among the apolar quadrics to f .
Varieties of sums of powers
The quadrics deﬁning an apolar elliptic quintic curve through an apolar subscheme of length 8, form an isotropic P4 for the corresponding quadratic form.
Varieties of sums of powers
Thus we look for a certain subvariety Y of the set of isotropic P4 ’s of a smooth quadric Q in P9 parametrizing apolar elliptic quintic curves.
Varieties of sums of powers
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