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Transformation Groups

, Volume 21, Issue 3, pp 593–618 | Cite as

ASSOCIATED FORMS OF BINARY QUARTICS AND TERNARY CUBICS

  • J. ALPER
  • A. V. ISAEV
  • N. G. KRUZHILIN
Article

Abstract

Let \( {\mathcal{Q}}_n^d \) be the vector space of forms of degree d ≥ 3 on ℂ n , with n ≥ 2. The object of our study is the map Φ, introduced in earlier articles by M. Eastwood and the first two authors, that assigns every nondegenerate form in \( {\mathcal{Q}}_n^d \) the so-called associated form, which is an element of \( {{\mathcal{Q}}_n^d}^{\left(d-2\right)*} \). We focus on two cases: those of binary quartics (n = 2, d = 4) and ternary cubics (n = 3, d = 3). In these situations the map Φ induces a rational equivariant involution on the projective space ℙ\( \left({\mathcal{Q}}_n^d\right) \), which is in fact the only nontrivial rational equivariant involution on ℙ\( \left({\mathcal{Q}}_n^d\right) \). In particular, there exists an equivariant involution on the space of elliptic curves with nonvanishing j-invariant. In the present paper, we give a simple interpretation of this involution in terms of projective duality. Furthermore, we express it via classical contravariants.

Keywords

Elliptic Curf Associate Form Zariski Open Subset Geometric Invariant Theory Projective Duality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityActonAustralia
  2. 2.Department of Complex AnalysisSteklov Mathematical InstituteMoscowRussia

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