 # ag.algebraic geometry – Most divisors on a round aren’t particular? Answer

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ag.algebraic geometry – Most divisors on a round aren’t particular?

I’ve a universal flush round $$C$$ of genus $$g$$ and stuck multiplicities $$a_1, dots, a_n geq 0$$ with $$sum a_i = g+1$$.

Q1 : For universal conspicuous factors $$p_1, dots, p_n in C$$, should $$sum a_i p_i$$ breathe a non-special divisor?

If $$n$$ is one, I simply need to keep away from the finitely many Weierstrass factors, and I’m on the lookout for an affinity of this for particular divisors. If the union of the helps of all particular divisors was finite, that’d breathe noble, however the areas $$G^r_d$$ can breathe higher dimension than the round. Even although I could make the uphold of the divisor universal, the multiplicities are fastened.

Q2 : For universal conspicuous factors and any $$b_1, dots, b_n$$, $$0 leq b_i < a_i$$, should $$sum b_i p_i$$ breathe non-special? What if $$sum b_i = g$$?

I used to be studying Akhil Matthew’s notes on a passage by Joe Harris, and the proof that there are finitely many Weierstrass factors is “something that eventually becomes obvious.” I’m attempting to generalize this, so I’d value being made cognizant if there’s a associated outcome. A paper casually claims that $$H^1(mathcal{O}(D)) = 0$$ for a universal divisor of diploma $$g$$ and I’m attempting to bridle it.

Apologies that I’m fairly inexperienced with curves.

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