# algebraic geometry – A question about a nondegenerate smooth curve in \$Bbb P^n\$ of degree \$n\$

Suppose that $$Xsubset Bbb P^n$$ is a nondegenerate smooth projective curve of degree $$n$$. Let $$H$$ be a hyperplane in $$Bbb P^n$$, $$D=text{div}(H)$$, and $$Qsubset |D|$$ be the linear subsystem of hyperplane divisors. Then we have the equalities $$n=dim (Q)leq dim |D|=dim L(D)-1leq n$$
Therefore $$Q=|D|$$ and $$dim L(D)=1+deg (D)$$. The latter equality implies that $$X$$ has genus zero, and the first equality implies that any divisor of degree $$n$$ is a hyperplane divisor.

This is a paragraph in p.217 of Miranda’s book Algebraic Curves and Riemann Surfaces, and I can’t understand the last sentence.

1. How do we know that the genus of $$X$$ is zero? I can only see that by Riemann-Roch $$g=dim L(K-D)$$.

2. How do we know that any divisor of degree $$n$$ is a hyperplane divisor? Is any positive divisor of degree $$n$$ contained in $$|D|$$?