# Finding polynomial equations for embedded Riemann surface

On the bottom of page 204 in Rick Miranda's book 'Algebraic curves and Riemann Surfaces' he writes "Since the hyperplane divisors on $$X$$ are exactly the divisors in the linear system |D|, we see that div$$(F_0) \sim kD$$ since $$F_0$$ has degree $$k$$."

The context is that $$X$$ is a compact Riemann surface embedded in $$\mathbb{P}^n$$ via the map $$x \mapsto [f_0(x) : \dots : f_n(x) ]$$ where $$f_0, \dots f_n$$ is a basis for $$L(D)$$, and $$F_0$$ is a homogeneous polynomial on $$\mathbb{P}^n$$ of degree $$k$$.

Does anyone understand why div$$(F_0) \sim kD$$? It is not clear to me what he means.

The divisors of any two nonzero homogeneous polynomials of the same degre on $$\Bbb P^n$$ are linearly equivalent since their difference is the divisor associated to the rational function obtained by taking the quotient of these two polynomials. So $$(F_0)\sim (h^k)$$ where $$h$$ is any linear form, and as $$(h^k)=k\cdot(h)$$ we have that $$(F_0)\sim kH$$ for $$H$$ a hyperplane divisor on $$\Bbb P^n$$. Now we restrict everything to your curve and maintain the equivalence (showing this last claim is probably a lemma or an exercise somewhere in your book, I've never used Miranda).