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We present some new applications of the theory of M-regularity ideated by Pareschi-Popa in a series of articles ranging
from 2002 to 2009. In particular:
(1) we will investigate the syzygies of (singular) Kummer varieties seeing when an power of an ample line bundle satisfies property N_p. (2) We will show that the 4-canonical map of a variety X of maximal Albanese dimension is always birational equivalent to its Iitaka fibration. If, furthermore, X is of general type, we will show that the 3-canonical map is birational (this is joint work with Jiang and Lahoz). (3) Finally we will present some evidence to a conjecture due to Pareschi, stating that the varieties of maximal Albanese dimension with Euler characteristic equal to 1 and Albanese image not fibered in tori are birational to products of theta divisors in pricipally polarized abelian varieties. |