Suppose $X$ is a non-singular irreducible projective variety over an algebraically closed field of characteristic zero. Let $L$ be an ample and globally generated line bundle on $X$. Then the complete linear system of $L$ gives a finite morphism (because of ampleness) to Projective space.
Can we find ample globally generated line bundles $L$ such that $h^0(X,L) = n+1$ where $n$ is the dimension of $X$?
We know that because $L$ is ample, $h^0(X,L)\geq n+1$. But in what situations/under what conditions does equality hold. In case of curves I can think of examples. Does this happen for higher dimensional varieties.