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Let $X$ be an abelian surface over complex numbers. Let $L$ be a totally symmetric line bundle of type $(r,r)$ for $r\geq 2$.

The involution $i$ on $X$ gives an action on $H=H^0(X,L)$ thus giving a splitting $$H=H+\oplus H-.$$

The dimensions of $H\pm$ is given by $\frac{r^2}{2}\pm 2.$

We have the linear systems $\mathbb{P}H\pm$. What is the base locus of these?

I think the base locus of $\mathbb{P}H+$ is empty. But the other is the 16 2-torsion points. Is this right? How to see it?

What do the linear systems corresponding to $H+$ and $H-$ describe? These are curves which are fixed by $i$. But do the curves in $H\pm$ differ in the way they pass through the 16 half-periods? What is the difference?

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    $\begingroup$ Locally near each fixed point the involution sends local coordinates $x,y$ to $-x,-y$ so that if $f(x,y)$ is a local expression for a section in $H-$, then it is odd and so has no constant term so the 16 2-torsion points are in the base locus. If $H$ is base point free (this is well studied in the litterature eg. Reider's theorem), then clearly the 2 torsion points are not in the base locus of $H+$. To check that no other point is in the base locus of I would look at the quotient variety $Y=X/Z_2$ and consider the corresponding line bundle say $M$ and use some version of Reider for sing vars. $\endgroup$
    – Hacon
    Commented Oct 16, 2016 at 15:50
  • $\begingroup$ @Hacon, thanks for the useful comment. Can you direct me to some reference? Also does it mean there are no sections in $H+$ which vanish at the 16 2-torsion points? Or could there be such curves? $\endgroup$
    – user52991
    Commented Oct 16, 2016 at 16:15
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    $\begingroup$ Suppose all sections of $H^0(L)^+$ vanish at a 2-torsion point, then since the same is true for sections of $H^0(L)^-$, we would have that all sections of $H^0(L)$ vanish at this point. The curves correspond to the zero sets of the sections. A good ref. is Lazarsfelds notes on linear series (in particular Thm 2.1) researchgate.net/publication/2218506_Lectures_on_Linear_Series $\endgroup$
    – Hacon
    Commented Oct 16, 2016 at 16:20
  • $\begingroup$ @Hacon, thank you I understand that. That is all sections of $H+$ cannot vanish at all 16 2-torsion points. But I was asking if there can exist some sections in $H+$ which vanish at these 16 points. $\endgroup$
    – user52991
    Commented Oct 16, 2016 at 16:44
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    $\begingroup$ Well, if $\dim H+\geq 2$, then pick $f_1,f_2\in H+$. If $f_i(x)\ne 0$, then let $f=f_2(x)f_1-f_1(x)f_2$. Clearly $f\in H+$ vanishes at $x$. Similarly if $\dim H+\geq 17$, then we can find $f\in H+$ vanishing at all 2-torsion points. If $\dim H+<17$ I am not sure. $\endgroup$
    – Hacon
    Commented Oct 16, 2016 at 23:54

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