On varieties with Ulrich twisted tangent bundles

Abstract

We study varieties $$X \subseteq {\mathbb {P}}^N$$ X ⊆ P N of dimension n such that $$T_X(k)$$ T X ( k ) is an Ulrich vector bundle for some $$k \in {\mathbb {Z}}$$ k ∈ Z . First we give a sharp bound for k in the case of curves. Then we show that $$k \le n+1$$ k ≤ n + 1 if $$2 \le n \le 12$$ 2 ≤ n ≤ 12 . We classify the pairs $$(X,{\mathcal {O}}_X(1))$$ ( X , O X ( 1 ) ) for $$k=1$$ k = 1 and we show that, for $$n \ge 4$$ n ≥ 4 , the case$$k=2$$ k = 2 does not occur.