# ag.algebraic geometry – Most divisors on a curve aren’t special?

I have a generic smooth curve $$C$$ of genus $$g$$ and fixed multiplicities $$a_1, dots, a_n geq 0$$ with $$sum a_i = g+1$$.

Q1 : For generic marked points $$p_1, dots, p_n in C$$, must $$sum a_i p_i$$ be a non-special divisor?

If $$n$$ is one, I just have to avoid the finitely many Weierstrass points, and I’m looking for an analogy of this for special divisors. If the union of the supports of all special divisors was finite, that’d be great, but the spaces $$G^r_d$$ can be greater dimension than the curve. Even though I can make the support of the divisor generic, the multiplicities are fixed.

Q2 : For generic marked points and any $$b_1, dots, b_n$$, $$0 leq b_i < a_i$$, must $$sum b_i p_i$$ be non-special? What if $$sum b_i = g$$?

I was reading Akhil Matthew’s notes on a course by Joe Harris, and the proof that there are finitely many Weierstrass points is “something that eventually becomes obvious.” I’m trying to generalize this, so I’d appreciate being made aware if there is a related result. A paper casually claims that $$H^1(mathcal{O}(D)) = 0$$ for a generic divisor of degree $$g$$ and I’m trying to check it.

Apologies that I’m quite inexperienced with curves.