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G. Pareschi and M. Popa give criterions for global generations andsurjectivity of multiplication maps of global sections of coherent sheaves onabelian varieties in the theory of M-regularity. In this paper, we generalizesome of their criterions via the M-regularity of $\mathbb{Q}$-twisted sheaves.As an application, we show that the M-regularity of a suitable$\mathbb{Q}$-twisted sheaf implies property $(N_p)$ and jet-ampleness for ampleline bundles on abelian varieties.

M-regularity of $\mathbb{Q}$-twisted sheaves and its application to linear systems on abelian varieties