Let $X\subset\mathbb{P}^N$ be a smooth projective variety of dimension $n$, say over $\mathbb{C}$. Assume that $H^i(X,\mathcal{O}_X)\neq 0$ for some $i$ with $0<i<n$. What is the minimum possible degree of $X$? Some elliptic scrolls have degree $2n+1$, can one do better?

3$\begingroup$ According to a result of KwakPark (Theorem A in arxiv.org/pdf/1510.03358.pdf) any bound excluding both elliptic scrolls and varieties with $H^{i}(\mathcal{O})=0$ for $0 < i < n$ has to be at least as sharp as $\deg(X) \geq N+3.$ $\endgroup$ – Yusuf Mustopa Nov 5 '17 at 21:00

2$\begingroup$ @Yusuf Mustopa: Great, thanks for the reference! Since $X\subset \mathbb{P}^N$ can be projected isomorphically to $\mathbb{P}^{2n+1}$, we get $\deg(X)\geq 2n+4$ unless $X$ is an elliptic scroll, which answers my question. If you want to put this as an answer I'll accept it. $\endgroup$ – abx Nov 6 '17 at 14:59
Proposition: Let $X$ be a smooth projective $n$dimensional ($n \geq 2$) nondegenerate subvariety of $\mathbb{P}^{N}$ satisfying the property that $H^{i}(\mathcal{O}_{X}) \neq 0$ for at least one $i \in \{1, \cdots ,n1\}.$ Then $${\rm deg}(X) \geq 2n+2.$$ If $X$ is not an elliptic scroll, then ${\rm deg}(X) \geq 2n+4.$
Proof: As abx observed, we may assume without loss of generality that $N=2n+1.$ By our hypothesis on the intermediate cohomology of $\mathcal{O}_{X}$ and Theorem A of [1], we have that ${\rm deg}(X) \geq 2n+2$, and that ${\rm deg}(X) \in \{2n+2,2n+3\}$ only if $X$ is an elliptic scroll. QED
[1] Kwak, Sijong and Park, Jinhyung, Geometric properties of projective manifolds of small degree, Math. Proc. Cambridge Philos. Soc. 160 (2016), no. 2, 257–277.