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On the rationality of the moduli schemes of vector bundles on curves. (English) Zbl 0920.14011

Let \(X\) be a smooth complete algebraic curve of genus \(g\geq 2\) over an algebraically closed field of characteristic zero. Let \(L\) be a line bundle on \(X\) of degree \(d.\) Consider the moduli scheme \(S_L(r,d)\) of stable vector bundles on \(X\) of rank \(r,\) with determinant \(L.\) \(S_L(r,d)\) is known to be smooth, irreducible and unirational. P. E. Newstead [Math. Ann. 215, 251-268 (1975; Zbl 0288.14003); correction: ibid. 249, 281-282 (1980; Zbl 0455.14003)] showed its rationality in several cases. The authors focus on the case in which \(d\) and \(r\) are relatively prime, which is known to coincide with \(S_L(r,d)\) being a fine moduli scheme. Let \(q'\) (respectively \(q''\)) be the class of \(d\) (respectively \(-d)\bmod r\), with \(r(g-1) \leq q'<rg\) (respectively \(r(g-1)\leq q''< rg\)). Let \(r'=rg- q'\) (respectively \(r''=rg-q''\)). In the paper under review, the authors prove the rationality of the fine moduli scheme \(S_L(r, d)\) assuming that \(r'\) and \(q'\) (or \(r''\) and \(q''\)) are relatively prime, under an additional numerical condition. Prompted by the referee to consider the work of H. U. Boden and K. Yokogawa [“Rationality of moduli spaces of parabolic bunbles”, http://xxx.lanl.gov/abs/alg-geom/9610013, see: J. Lond. Math. Soc. 59, 461-478 (1999)], the authors point out that the additional numerical condition can be removed.
A variety \(W\) is called stably rational of level less than or equal to \(w\) if its product with a projective space of dimension \(w\) is rational. E. Ballico previously proved that \(S_L(r,d)\) is stably rational [J. Lond. Math. Soc., II. Ser. 30, 21-26 (1984; Zbl 0512.14032)]. The proof of the main result of the paper under review relies on obtaining an upper bound for the level of stable rationality for \(S_L(r,d).\) The authors obtain a quadratic bound in the rank, while Boden and Yokogawa give a linear bound in the rank. The work of D. Butler [“On the rationality of \(SU(r,d)\)”, http://xxx.lanl.gov/abs/alg-geom/9705008] obtains further results on the rationality of the moduli scheme.

MSC:

14H60 Vector bundles on curves and their moduli
14M20 Rational and unirational varieties
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References:

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