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Associated forms of binary quartics and ternary cubics. (English) Zbl 1372.14038
The main object of the paper under review is the complex vector space \(Q_ n^ d\) of \(n\)-ary forms of degree \(d\) with complex coefficients (\(n\geq 2\), \(d\geq 3\)). More precisely, the authors are interested in the map \(\Phi\) sending a nondegenerate \(n\)-ary \(d\)-form \(f\) to the associated form which is an element of the space dual to \(Q_ n ^{n(d-2)}\). Their focus is on the cases of binary quartics and ternary cubics, the only cases when \(\Phi\) preserves the degree. They show that in each of these cases the projectivization of \(\Phi\) induces an equivariant (with respect to an action of \(SL_n\)) involution on the projectivization of the space of nondegenerate forms with one orbit removed. Furthermore, they show that a such a nontrivial rational equivariant involution is unique. In particular, \(\Phi\) yields a unique equivariant involution on the space of elliptic curves with nonvanishing \(j\)-invariant. They give a simple interpretation of this involution in terms of projective duality and express it via classical contravariants, in the spirit of Cayley and Sylvester.
Eventual applications in singularity theory are also explained.

MSC:
14L24 Geometric invariant theory
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
14H52 Elliptic curves
32S10 Invariants of analytic local rings
32S25 Complex surface and hypersurface singularities
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