Algebraic Geometry notation of poles

Question

Let $X$ be a scheme (for instance an algebraic curve) over $Spec(k)$ being $k$ an algebraically closed field of characteristic $0$. Let $p\in X$ be a closed point, and let $E$ be a vector bundle of rank $n$ over $X$, or equivalently, a locally free sheaf of $\mathcal{O}_X$-modules of rank $n$, which will be denoted by $\mathcal{E}$ (following Hartshorne notation). What does the notation $E(-mp)$ means? This should be a locally free sheaf because at some point I have seen the notation $H^0(X,E(-mp))$, and $H^1(X,E(-mp))$. Someone can help me?

Answer 1

The notation $E(-mp)$ means $E\otimes_{\mathcal{O}_X} \mathcal{O}_X(-mp)$, where the latter sheaf is the line bundle associated to the divisor $-mp$. Context clues suggest that $X$ is a curve, $p$ is a point, and $m$ is an integer - then $\mathcal{O}_X(-mp)$ can be considered as the sheaf of rational functions which are regular outside of $p$ and have poles of at most order $m$ at $p$.

For more information on invertible sheaves associated to divisors, see Hartshorne chapter II section 6, Vakil chapter 14, or Stacks 0C4S.