It is known that in characteristic zero, in a linear system (not necessarily complete) in $\mathbb{P}^3$ a general element is irreducible. Now my question is the following:
Suppsoe we have a subspace $V \subset H^0(\mathbb{P}^3, \mathcal{O}(3))$ of dimension $\ge 3$. Then is it possible  always to  find an element in $V$ which is reducible ?