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Linear stability and stability of syzygy bundles. (English) Zbl 1403.14019

D. Mumford [in: Algebraic geometry. The Johns Hopkins centen. Lect., Symp. Baltimore/Maryland 1976, 6–8 (1977; Zbl 0497.14004)] introduced the notion of linear stability for projective varieties. In [Int. J. Math. 23, No. 12, Paper No. 1250121, 25 p. (2012; Zbl 1278.14050)], E. C. Mistretta and L. Stoppino generalized this notion to linear series over a curve. If \(L\) is a generated line buldle on a smooth irreducible projective curve \(C\) , \(V\subseteq H^0(L)\) a linear subspace that generates \(L\) and \(M_V,_L\) the kernel of the evaluation map \(V\otimes O_C\rightarrow L\), then in [J. Lond. Math. Soc., II. Ser. 78, No. 1, 172–182 (2008; Zbl 1146.14022)] E. C. Mistretta showed that the linear stability of \((L,V)\) is equivalent to the stability of \(M_V,_L\) under some conditions on the genus of \(C\). In the present paper, the authors give a positive answer to a conjecture of Mistretta and Stoppino for \(k\)-gonal curves. They give conditions under which the stability of \(M_V,_L\) is equivalent to the linear stability of \((L,V)\). They give a criterion where this equivalence is satisfied on \(k\)-gonal curves. [L. Ein and R. Lazarsfeld, Lond. Math. Soc. Lect. Note Ser. 179, 149–156 (1992; Zbl 0768.14012)].

MSC:

14C20 Divisors, linear systems, invertible sheaves
14H10 Families, moduli of curves (algebraic)
14H51 Special divisors on curves (gonality, Brill-Noether theory)
14H60 Vector bundles on curves and their moduli
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