Abstract
Using the connection discovered by Hassett between the Noether-Lefschetz moduli space mathcal {C}_{42} of special cubic fourfolds of discriminant 42 and the moduli space mathcal {F}_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over mathcal {F}_{22} is unirational.
Highlights
The 19-dimensional moduli space Fg of polarized K 3 surfaces of genus g, parametrizing pairs [S, H ], where S is a K 3 surface and H ∈ Pic(S) is a primitive polarization class satisfying H 2 = 2g − 2, is one of the most intriguing parameter spaces in algebraic geometry
On the other hand, using vector bundles on various rational homogeneous varieties, in a celebrated series of papers Mukai [20,21,22,23,24] described the construction of general polarized K 3 surfaces of genus g ≤ 12, as well as for g = 13, 16, 18, 20
Nuer [29] first showed that F14 is uniruled. This was improved in [9], where we showed that the universal K 3 surface F14,1 is rational, F14 is unirational
Summary
Let Hyp be the moduli space of pairs [ , L], where is an integral 8nodal curve of arithmetic genus 12, whose normalization ν : C → is a hyperelliptic curve of genus 4 and L ∈ W82( ), that is, L is a line bundle of degree 8 on with h0( , L) ≥ 3. Via the isomorphism T ∼= Bl9(P2), one realizes as an octic plane curve with 17 nodes divided in two groups: namely the 9 points where P2 is blown up and the remaining 8 nodes This plane model is helpful to prove the result: Theorem 1.4 The moduli space Hyp is unirational. First we show that Hyp is unirational (see Theorem 3.5), using the explicit unirational parametrization found in this way, we show that the map χ : P Hyp is well defined, as well as birational
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