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The rationality of some moduli spaces of plane curves. (English) Zbl 0661.14022

The author shows that the moduli space \({\mathbb{P}}(H^ 0({\mathcal O}_{{\mathbb{P}}^ 2_{{\mathbb{C}}}}(d)))/SL_ 3\) for plane curves of degree d is rational, under the condition that \(d\equiv 1\quad (mod\quad 9)\) and \(d\geq 19\), or \(d\equiv 1\quad (mod\quad 4),\) by analyzing explicitly the quotient space. He also shows the rationality of the quotient space of the space of pencils of binary forms of degree 2k with \(k\geq 5\) by \(PGL_ 2\), and the rationality of the moduli space for polarized K3 surfaces of degree 18 by using a result of S. Mukai’s unpublished paper [“Curves, K3 surfaces and Fano threefolds that are complete intersections in homogeneous spaces”].
Reviewer: T.Sekiguchi

MSC:

14H10 Families, moduli of curves (algebraic)
14M20 Rational and unirational varieties
14J10 Families, moduli, classification: algebraic theory
14D20 Algebraic moduli problems, moduli of vector bundles
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References:

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