Linear stability and rank-two Clifford indices of algebraic curves, with applications

Abstract

We prove that any vector bundle computing the rank-two Clifford index of a smooth projective algebraic curve is linearly semistable. We also identify conditions under which such bundles become linearly stable, thereby addressing a question posed by Castorena, Hitching, and Luna [ Linear stability of coherent systems and applications to Butler’s conjecture , Preprint (2023), arXiv:2312.09309] in the rank-two case. Furthermore, we demonstrate that, in certain special cases, this property is equivalent to the (semi)stability of the associated Lazarsfeld–Mukai bundles. This yields a positive solution, in specific cases, to a generalized version of a conjecture proposed by Mistretta and Stoppino [ Linear series on curves: stability and Clifford index , Internat. J. Math. 23 (2012), 1250121]. We also study the moduli space

of generated

alpha α $\alpha$

-stable coherent systems of type ( n , d , 5) for small values of

alpha α $\alpha$

and

n equals 2 comma 3 n = 2 , 3 $n=2,3$

. We show that a general element of an irreducible component

upper X subset of or equal to upper S 0 left parenthesis 2 comma d comma 5 right parenthesis X ⊆ S 0 ( 2 , d , 5 ) $X \subseteq S_0(2, d, 5)$

or

upper X subset of or equal to upper S 0 left parenthesis 3 comma d comma 5 right parenthesis X ⊆ S 0 ( 3 , d , 5 ) $X \subseteq S_0(3, d, 5)$

is linearly stable whenever

2 delta 2 less than or slanted equals d less than or slanted equals 3 g divided by 2 2 δ 2 ⩽ d ⩽ 3 g / 2 $2\delta_2\leqslant d\leqslant 3g/2$

. As an application of this, we prove that Butler’s conjecture holds non-trivially for

upper S 0 left parenthesis 2 comma d comma 5 right parenthesis S 0 ( 2 , d , 5 ) $S_0(2,d,5)$

within the given range for d .