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The rationality of the moduli space of one-pointed ineffective spin hyperelliptic curves via an almost del Pezzo threefold. (English) Zbl 1408.14097

A genus \(g\) hyperelliptic ineffective spin curve \((C, \theta)\) consists of a genus \(g\) hyperelliptic curve \(C\) and a half canonical divisor \(\theta\) on \(C\) with \(|\theta|\) empty.
This paper investigates the moduli space \(\mathcal{S}^{0, \text{hyp}}_{g,1}\) of one-pointed genus \(g\) hyperelliptic ineffective spin curves, and show that it is an irreducible rational variety. As a corollary, the moduli space \(\mathcal{S}^{0, \text{hyp}}_{g}\) of genus \(g\) hyperelliptic ineffective spin curves is an irreducible unirational variety.
The method is to consider an almost del Pezzo \(3\)-fold \(B_a\), which is a small resolution of a degeneration of the quintic del Pezzo threefold. By studying the geometry of \(B_a\), \(\mathcal{S}^{0, \text{hyp}}_{g,1}\) can be realized (birationally) as a quotient of family of rational curves on \(B_a\), and the rationality can be solved by computing invariants.

MSC:

14H10 Families, moduli of curves (algebraic)
14E30 Minimal model program (Mori theory, extremal rays)
14J45 Fano varieties
14N05 Projective techniques in algebraic geometry
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