On the rationality of the moduli schemes of vector bundles on curves
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. \textit{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 \textit{H. U. Boden} and \textit{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. \textit{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 \textit{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.
scientific article; zbMATH DE number 1089767
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