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A computational solution to a question by Beauville on the invariants of the binary quintic. (English) Zbl 1102.14024
The author considers the natural map \(S \to G_S\) from an element \(S\) in the moduli space of quartic del Pezzo surfaces, to the set of conjugacy classes of subgroups (in the Cremona group \(\text{Cr}(\mathbb P^2)\)) isomorphic to \((\mathbb Z/(4))^4\). The map is proved to be injective by A. Beauville [arXiv:math.AG/0502123]. The author generalizes this result. He uses Maple and proves: For a set \(\Lambda\) of five distinct unordered points in \(\mathbb P^1\), a certain quintic polynomial \(R_{\Lambda}\) constructed from certain \(j\)-invariants, uniquely determines the SL\(_2\) orbit of \(\Lambda\).

14J26 Rational and ruled surfaces
Full Text: DOI
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