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?



Minimum degree of a variety with $H^i(X,\mathcal{O}_X)\neq 0$ for some $i$ with $0<i<\dim(X)$
**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 $i \in \{1, \cdots ,n-1\}.$ 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. 


