Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion line bundle. Therefore, we have that $deg(A)=deg(A\otimes M)$. What is $h^0(C,A\otimes M)$ in general?

More precisely,

We have the subvariety $W^r_d(C)=\{A\in Pic^d(C)|h^0(C,A)\geq r+1\}$ of $Pic^d(C)$. A 2-torsion line $M$ acts on $Pic^d(C)$. Assume $0< d < g$ Then what will the action of $M$ on $Pic^d(C)$ do to $W^r_d(C)\setminus W^{r+1}_d(C)$ ? In fact consider $W^0_d(C)=\{A\in Pic^d(C):h^0(C,A)\geq 1\}$. This has expected dimension $\rho(g,0,d)<g$ when $d<g$. This will get translated to some other variety under translation by $M$ right? So that $h^0(A\otimes M)=0$. How do we prove this?

I have attempted the following: Consider $t_M:Pic^d(C)\longrightarrow Pic^d(C)$ given by $B\mapsto B\otimes M$. This is an isomorphism. Now $A\in Pic^d(C)$ is such that $h^0(A)\geq 1$. That is $A\in W^0_d(C)$.We have that $W^0_d(C)\subset Pic^d(C)$. Now, every component of $W^0_d(C)$ has expected dimension (when $d<g$): $\rho(g,0,d)=g-1(g-d+0)=d< g$.\ Now $Pic^d(C)$ is a $g$-dimensional variety, so under translation by $M\in Pic^0(C)$, a general element of the subvariety $W^0_d(C)$ of $Pic^d(C)$ moves to another outside $W^0_d(C)$. Hence for a general $A\in W^0_d(C)$, $h^0(C,A\otimes M)=0$.

Is this correct? How to improve upon this answer?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion line bundle. Therefore, we have that $deg(A)=deg(A\otimes M)$. What is $h^0(C,A\otimes M)$ in general? Can we say that it is zero?

More precisely,

We have the subvariety $W^r_d(C)=\{A\in Pic^d(C)|h^0(C,A)\geq r+1\}$ of $Pic^d(C)$. A 2-torsion line $M$ acts on $Pic^d(C)$. Assume $0< d < g$ Then what will the action of $M$ on $Pic^d(C)$ do to $W^r_d(C)\setminus W^{r+1}_d(C)$ ? In fact consider $W^0_d(C)=\{A\in Pic^d(C):h^0(C,A)\geq 1\}$. This has expected dimension $\rho(g,0,d)<g$ when $d<g$. This will get translated to some other variety under translation by $M$ right? So that $h^0(A\otimes M)=0$. How do we prove this?

I have attempted the following: Consider $t_M:Pic^d(C)\longrightarrow Pic^d(C)$ given by $B\mapsto B\otimes M$. This is an isomorphism. Now $A\in Pic^d(C)$ is such that $h^0(A)\geq 1$. That is $A\in W^0_d(C)$.We have that $W^0_d(C)\subset Pic^d(C)$. Now, every component of $W^0_d(C)$ has expected dimension (when $d<g$): $\rho(g,0,d)=g-1(g-d+0)=d< g$.\ Now $Pic^d(C)$ is a $g$-dimensional variety, so under translation by $M\in Pic^0(C)$, a general element of the subvariety $W^0_d(C)$ of $Pic^d(C)$ moves to another outside $W^0_d(C)$. Hence for a general $A\in W^0_d(C)$, $h^0(C,A\otimes M)=0$.

Is this correct? How to improve upon this answer?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion line bundle. Therefore, we have that $deg(A)=deg(A\otimes M)$. What is $h^0(C,A\otimes M)$ in general?

More precisely,

We have the subvariety $W^r_d(C)=\{A\in Pic^d(C)|h^0(C,A)\geq r+1\}$ of $Pic^d(C)$. A 2-torsion line $M$ acts on $Pic^d(C)$. Assume $0< d < g+r$ Then what will the action of $M$ on $Pic^d(C)$ do to $W^r_d(C)\setminus W^{r+1}_d(C)$ ? In fact consider $W^0_d(C)=\{A\in Pic^d(C):h^0(C,A)\geq 1\}$. This has expected dimension $\rho(g,0,d)<g$ when $d<g$. This will get translated to some other variety under translation by $M$ right? So that $h^0(A\otimes M)=0$. How do we prove this?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion line bundle. Therefore, we have that $deg(A)=deg(A\otimes M)$. What is $h^0(C,A\otimes M)$ in general?

More precisely,

We have the subvariety $W^r_d(C)=\{A\in Pic^d(C)|h^0(C,A)\geq r+1\}$ of $Pic^d(C)$. A 2-torsion line $M$ acts on $Pic^d(C)$. Assume $0< d < g$ Then what will the action of $M$ on $Pic^d(C)$ do to $W^r_d(C)\setminus W^{r+1}_d(C)$ ? In fact consider $W^0_d(C)=\{A\in Pic^d(C):h^0(C,A)\geq 1\}$. This has expected dimension $\rho(g,0,d)<g$ when $d<g$. This will get translated to some other variety under translation by $M$ right? So that $h^0(A\otimes M)=0$. How do we prove this?

I have attempted the following: Consider $t_M:Pic^d(C)\longrightarrow Pic^d(C)$ given by $B\mapsto B\otimes M$. This is an isomorphism. Now $A\in Pic^d(C)$ is such that $h^0(A)\geq 1$. That is $A\in W^0_d(C)$.We have that $W^0_d(C)\subset Pic^d(C)$. Now, every component of $W^0_d(C)$ has expected dimension (when $d<g$): $\rho(g,0,d)=g-1(g-d+0)=d< g$.\ Now $Pic^d(C)$ is a $g$-dimensional variety, so under translation by $M\in Pic^0(C)$, a general element of the subvariety $W^0_d(C)$ of $Pic^d(C)$ moves to another outside $W^0_d(C)$. Hence for a general $A\in W^0_d(C)$, $h^0(C,A\otimes M)=0$.

Is this correct? How to improve upon this answer?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion line bundle. Therefore, we have that $deg(A)=deg(A\otimes M)$. What is $h^0(C,A\otimes M)$ in general?

More precisely,

We have the subvariety $W^r_d(C)=\{A\in Pic^d(C)|h^0(C,A)\geq r+1\}$ of $Pic^d(C)$. A 2-torsion line $M$ acts on $Pic^d(C)$. Assume $0< d<g+r$, and that $W^r_d(C)\setminus W^{r+1}_d(C)=\{A\in Pic^d(C)|h^0(C,A)=r+1\}$ is nonempty. Then what will the action of $M$ on $Pic^d(C)$ so to $W^r_d(C)\setminus W^{r+1}_d(C)$ ? In fact consider $W^0_d(C)=\{A\in Pic^d(C):h^0(C,A)\geq 0\}$. This will get translated to some other variety under translation by $M$ right? So that $h^0(A\otimes M)=0$. How do we prove this?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion line bundle. Therefore, we have that $deg(A)=deg(A\otimes M)$. What is $h^0(C,A\otimes M)$ in general?

More precisely,

We have the subvariety $W^r_d(C)=\{A\in Pic^d(C)|h^0(C,A)\geq r+1\}$ of $Pic^d(C)$. A 2-torsion line $M$ acts on $Pic^d(C)$. Assume $0< d < g+r$ Then what will the action of $M$ on $Pic^d(C)$ do to $W^r_d(C)\setminus W^{r+1}_d(C)$ ? In fact consider $W^0_d(C)=\{A\in Pic^d(C):h^0(C,A)\geq 1\}$. This has expected dimension $\rho(g,0,d)<g$ when $d<g$. This will get translated to some other variety under translation by $M$ right? So that $h^0(A\otimes M)=0$. How do we prove this?