Let $Y\subseteq \mathbb{P}^n$ be the zero locus of $f_1,...,f_k$ of degree $d_1,...,d_k$, and put $d=\sum d_i$. If $Y$ is nonsingular and a complete intersection, then $dim(H^0(\omega_Y))=dim(H^0(\mathcal{O}(d-n-1)))$, and hence we can compute its geometric genus.
I'm wondering about the case where $Y$ is not necessarily a complete intersection, but is still nonsingular. For instance if $Y$ is not codimension $k$ (ie the twisted cubic). Assuming we don't have any redundant $f_i$, is there anything we can say about the geometric genus? Even rough information would be helpful, ie a lower/upper bound, perhaps involving the difference between $codim(Y)$ and $k$.
Thanks