Serre's vanishing theorem (SV) states that, on a projective variety $X$ with a choice of ample line bundle $\mathcal{O}\_X(1)$, for any coherent sheaf $F$, we have
$$H^i(X,F(m))=0,\quad m>>0$$
for every $i>0$ (i.e. there's an $m$ big enough such that the vanishing occurs for any bigger multiple $m'>m$ of the polarization and for every index $i$), where $F(m)$ denotes the twist $F\otimes\mathcal{O}\_X(m)$.


If $D\_\*:=\{ D\_m\}\_{m\in\mathbb{N}}$ is a sequence of divisors on the projective variety $X$, we say that $D\_\*$ *satisfies SV on* $X$ if, for every coherent sheaf $F$ on $X$,
$$H^i(X,F(D\_m))=0,\quad m>>0$$
for every $i>0$, where as usual $F(D\_m)=F\otimes\mathcal{O}\_X(D\_m)$.


I'm mostly interested in the case in which:


($\star$) $X$ is a normal surface over $\mathbb{C}$.



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> **Q1.** Is there some "general" and "nontrivial" criterion for a sequence $D\_\*$ to satisfy SV? E.g. $D\_\*:=$ a subsequence of $\{m\cdot D\_0\}$ for a fixed ample $D\_0$ clearly satisfies it, but it's a trivial consequence of the usual SV.
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> Also, is there some such property that is valid for every $X$? (For example, could it be something like: $D\_\*$ eventually forms a subsequence of a sequence of ample divisors which is "increasingly positive" in some specified sense?)
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Narrowing down the question (and assuming ($\star$), if it is of any use):



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> **Q2.** Assume that $D\_\*$ is a sequence of (not necessarily effective) *ample* divisors such that $D\_{m+1}-D\_m$ is effective. Does $D\_\*$ satisfy SV?
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Sequences of divisors satisfying Serre vanishing?
Claim: $D\_\*$ is SV iff for any ample divisor $H$ there exists $m(H)$ such that if $m\geq m(H)$, then $D\_m-H$ is nef.


Suppose $D\_\*$ is SV. Fix $A$ very ample. Since $D\_\*$ is SV, then $H^i(D\_m-H-jA)=0$ for all $i>0$ and $0\leq j\leq \dim X +1$. By Castelnuovo-Mumford regularity $D\_m-H$ is generated by global sections and in particular nef.


Suppose that for any ample divisor $H$ there exists $m(H)$ such that if $m\geq m(H)$, then $D\_m-H$ is nef. Let $F$ be a coherent sheaf on $X$ and $A$ an ample line bundle on $X$, then by Fujita vanishing there exists an integer $t\_0$ such that $H^i(F(tA+P))=0$ for any $t\geq t\_0$ and $P$ a nef line bundle. But then, for any $m\geq m(t\_0A)$, we have $D\_m-t\_0A$ is nef and hence $H^i(F(D\_m))=0$.


