Timeline for Canonical divisor of a curve base point free (if g>0)
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2011 at 23:08 | comment | added | GiulioP | I think I found the solution: if $P$ is a place as in the equival. statement of my problem as above, suppose that $l(P)=\deg(P)+1$ (equiv. such a differential does not exist). Then there exists $z$ trascendental over $K$ whose pole divisor is $P$. Multiply a non-zero holomorphic differential ($g\geq0$) by a sufficiently large power of $z$ we get an holom. differ. whose associated divisor does not contain $P$. | |
| Nov 15, 2011 at 17:21 | comment | added | GiulioP | exactly, that is what I meant... I used clifford´s theorem (as in the exercises of Stichtenoth book) to solve the problem, but I would like a simple proof which avoids that. Essentially is the fact that if $l(A)=\deg(A)+1$ in genus greater than zero then $A$ is principal, which I can´t prove. Is it possible to prove it without extending the base field to the algebraic closure, as it seems that you suggest? It seems that the result is contained in the book of Deuring, "Lectures in the theory of alf. func.", Lemma Lectures 10, §20, but I don´t understand the proof... | |
| Nov 15, 2011 at 16:42 | comment | added | rita | In the algebraically closed case (that is enough for your purpose, as remarked above) it follows directly by Riemann-Roch, without using Clifford's theorem. | |
| Nov 15, 2011 at 16:38 | comment | added | Daniel Loughran | Surely the case where $K$ is not algebraically closed follows from the case where $K$ is algebraically closed? | |
| Nov 15, 2011 at 16:23 | comment | added | GiulioP | if $A$ is a divisor of $F/K$ such that $0\leq\deg(A)\leq 2g-2$, then $$l(A)\leq 1+\deg(A)/2$$ where as usual $l(A)$ is the dimension over $K$ of the vector space $L(A)=\{x\in F|(x)+A\geq0\}$. | |
| Nov 15, 2011 at 15:47 | comment | added | Damian Rössler | Could you remind me (us) what Clifford's theorem says ? | |
| Nov 15, 2011 at 15:42 | history | asked | GiulioP | CC BY-SA 3.0 |