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Some special families of hyperelliptic curves

Tanush Shaska

I-AIMS paper 2003-2 · Published in J. Algebra Appl. 3 (2004), 75–89 · 2004

Let ℒ_gᴳ denote the locus of hyperelliptic curves of genus g whose automorphism group contains a subgroup isomorphic to G. We study spaces ℒ_gᴳ for G ≅ ℤₙ, ℤ₂ ⊕ ℤₙ, ℤ₂ ⊕ A₄, or SL₂(3). We show that for G ≅ ℤₙ, ℤ₂ ⊕ ℤₙ, the space ℒ_gᴳ is a rational variety and find generators of its function field. For G ≅ ℤ₂ ⊕ A₄, SL₂(3) we find a necessary condition in terms of the coefficients, whether or not the curve belongs to ℒ_gᴳ. Further, we describe algebraically the loci of such curves for g ≤ 12 and show that for all curves in these loci the field of moduli is a field of definition.

MSC classes: 14Q05, 14Q15, 14R20, 14D22

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References

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