n-Dimensional Fano varieties of wild representation type

The aim of this work is to contribute to the classification of projective varieties according to their representation type, providing examples of n  -dimensional varieties of wild representation type, for arbitrary n⩾2n⩾2. More precisely, we prove that all Fano blow-ups of PnPn at a finite number of points are of wild representation type exhibiting families of dimension of order r2r2 of simple (hence, indecomposable) ACM rank r   vector bundles for any r⩾nr⩾n. In the two dimensional case, the vector bundles that we construct are moreover Ulrich bundles and μ-stable with respect to certain ample divisor.