Existence of $g_{d}^{r}$ implies existence of which $g_{d'}^{r'}$'s?

Question

Suppose I have a Riemann surface with a $g_{d}^{r}$. I am wondering what $g_{d'}^{r'}$'s exist for sure. For instance, since $h^{0}\left(L\left(-p \right)\right)$ is either $h^{0}\left(L\right)$ in case $p$ is a base point or $h^{0}\left(L\right)-1$ otherwise, the existence of a $g_{d}^{r}$ implies the existence of a $g_{d-1}^{r-1}$. Another tool is to consider a complete linear system and then consider the dual one using Riemann Roch theorem.

I was wondering whether there are other techniques we can use. In addition, I am also interested in the base-point-free case. For instance, given a base-point-free $g_{d}^{r}$, which linear system I can "generate" from it are still base-point-free?

Answer 1

Your question has to do with Brill-Noether Theory.

Fix $C$ a compact Riemann surface of genus $g$, and also fix $r,d$.

Question. Does $C$ have a $g^r_d$? If yes, how many?

To answer, you can define the Brill-Noether number $$\rho=\rho(g,r,d)=(r+1)(d-r)-rg$$ and the theorem you need is the following:

Theorem. There is a projective moduli scheme $G^r_d(C)$ parameterizing $g^r_d$'s on $C$. Each component has dimension $\geq\rho$. Moreover, for general $C$, one has $\dim G^r_d(C)=\rho$, with $G^r_d(C)=\emptyset$ if $\rho<0$.

Note that if $\rho<0$ (and $C$ is not general), anything can happen: $G^r_d(C)$ might or might not be empty. However, if $\rho\geq 0$, then every compact Riemann surface $C$ gives a non-empty moduli space $G^r_d(C)$.

So, knowing that $(g,r,d)$ gives a non-negative $\rho$, your new triple $(g,r',d')$ should give you a non-negative $\rho'$ as well, in order to be sure that a $g^{r'}_{d'}$ exists. I do not know if one can do much better than this.