On the rationality of the moduli schemes of vector bundles on curves
type: Item
instance of: scholarly article
title: On the rationality of the moduli schemes of vector bundles on curves
zbMATH Open document ID: 0920.14011
author: E. Ballico
author: Barbara Russo
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1998-01-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108427
review text: 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 r…
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 1089767
zbMATH Keywords: moduli of vector bundles
zbMATH Keywords: rationality of the fine moduli scheme
reviewed by: Gian Mario Besana
MaRDI profile type: Publication
cites work: Stable Rationality for the Variety of Vector Bundles over an Algebraic Curve
cites work: Stably rational non rational varieties
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cites work: Q3808217
cites work: On Newstead's conjecture on vector bundles on algebraic curves
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cites work: Periods of a Moduli Space of Bundles on Curves
cites work: Correction: ''Rationally of moduli spaces of stable bundles''
cites work: The moduli space of vector bundles over an algebraic curve
cites work: Q3277837
cites work: Q3665240
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description: scientific article; zbMATH DE number 1089767