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 ${\rm deg}(X) \in \{2n+2,2n+3\}$ if and 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.

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.

Proposition: Let $X$ be a smooth projective $n-$dimensional 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 ${\rm deg}(X) \in \{2n+2,2n+3\}$ if and 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.

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 ${\rm deg}(X) \in \{2n+2,2n+3\}$ if and 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.