On existence of Ulrich sheaf

Let X be a smooth projective variety carrying an Ulrich bundle. In the first part of this note, we construct an Ulrich sheaf on n-th symmetric power of X, which is a singular variety when $DimX >1$. As a consequence, we get the existence of an Ulrich bundle on Hillb^{n}C, where C is a smooth projective curve. Let A be an abelian variety which carries an Ulrich bundle. In the second part of this note, we show the existence of Ulrich bundle on the blow up of $A \times A$ along $A \times \{0\}$.