# Effective divisor on smooth projective variety

algebraic-geometrydivisors-algebraic-geometry

Let $$X$$ be a smooth projective variety and $$D_0$$ be a divisor on $$X$$ (over the algebraically closed field $$K$$). Then according to Harthshorne Proposition $$7.7(b)$$(Chapter $$2$$) every effective divisor on $$X$$ is of the form $$(s)_0$$ (i.e. zero of $$s$$) for some $$s \in H^0(L(D_0))$$.

In this context I have $$2$$ questions : $$(i)$$ What if the line bundle associated to $$D_0$$ does not have any non-zero global section? (Maybe this can't happen follows from something obvious which I am unable to see)

$$(ii)$$ Can we conclude from above that every nonzero effective divisor on $$X$$ always posses atleast $$1$$ non-zero global section?

Can someone briefly indicate the chain of argument for $$(ii)$$.

For $$(ii)$$ can we argue as follows :if $$E$$ is an effective divisor then it's linearly equivalent to an $$(s)_0$$ for some non-zero $$s \in H^0(L(D))$$ for some divisor $$D$$ on $$X$$. Then $$L(E) \cong L((s)_0) \cong L(D)$$ and hence $$E$$ has a section. Please correct me if this is wrong

(i) If $$h^0(X,\mathcal{O}(D_0)) = 0$$, then $$D_0$$ cannot be effective by your cited result. Namely if $$D_0$$ were effective, then there exists $$0 \neq s \in \Gamma(X, \mathcal{O}(D_0))$$ such that $$D_0 \sim (s)_0$$ and in particular $$\Gamma(X, \mathcal{O}(D_0)) \neq 0$$.
(ii) If $$D$$ is effective, then it has non-trivial global sections. This is the contrapositive of (i).
Indeed, Hartshorne also notes immediately after that proposition the following statement: Let $$D_0$$ be an effective divisor, then $$|D_0| \cong (\Gamma(X, \mathcal{O}(D_0)) - \{0 \})/k^{\times}$$ where $$|D_0|$$ denotes the complete linear system of $$D_0$$. Non-trivial global sections up to $$k^{\times}$$ correspond to the realization of $$D_0$$.
In general on nice enough schemes, invertible sheaves $$\mathcal{L}$$ admit non-zero rational sections $$s$$ such that $$\mathcal{L} \cong \mathcal{O}(\operatorname{div}{s})$$, Vakil 14.2.E. The sheaves corresponding to effective divisors are precisely the ones which admit such non-zero global sections.