The Stratified and the Strong Maximal Rank Conjecture in $\mathbb{P}^3$ and $\mathbb{P}^4$

We prove that the Strong Maximal Rank Conjecture holds for quadrics in $\mathbb{P}^3$ and we prove the existence of a component of the expected dimension in $\mathbb{P}^4$, as well as in a wide range of parameters $(g,d)$ in $\mathbb{P}^r$ with $r\ge 5$. We propose the Stratified Strong Maximal Rank Conjecture which also takes into account the rank $k$ of quadrics and prove it works in most of the cases when $k=3$ and $k=4$. We also prove a partial result that concerns the unrepresentability of the canonical bundle of a general curve as a sum of $3$ pencils.