Let $X$ be an abelian surface over $\mathbb{C}$. Consider the Kummer surface $K$ associated to $X$, that is the quotient of $X$ by the action of involution on $X$, $x\mapsto -x$. Kummer surface is a singular surface with 16 nodes. Does this mean that every curve through the 16 nodes is a singular curve? If we start with a smooth curve on a the abelian surface through the sixteen 2-torsion points, is it not isomorphic to its image? Thanks!



Curve through the 16 singular points of a Kummer surface
Regarding your first question, the answer is *no*. 


Following J. Silverman's suggestion, let me provide an example of an abelian surface $A$ and a smooth curve $C$, fixed by the involution $(-1)\_A$, and containing all the $2$-torsion points of $A$. Therefore, the image $C'$ of $C$ in $K := \textrm{Kum}(A)$ will be a smooth curve through the $16$ nodes of $K$.


Let $(A, \, \mathcal{L})$ be a $(1, \, 2)$-polarized abelian surface. We can assume with no loss of generality that $\mathcal{L}$ is a symmetric polarization, i.e. that $(-1)\_A^\* \mathcal{L} = \mathcal{L}$. Then it is possible to show what follows:


* $\dim H^0(A, \, \mathcal{L}^2)=8;$
* $\dim H^0(A, \, \mathcal{L}^2)^+=6\,$ and $\, \dim H^0(A, \, \mathcal{L}^2)^-=2,$ 
where "+" and "-" denote the space of $(-1)\_A$-invariant and $(-1)\_A$-anti-invariant sections, respectively;
* the pencil $|\mathcal{L}^2|^-$ of anti-invariant sections has $16$ base points, namely the $16$ points of 2-torsion of $A$, and the general element of this pencil is smooth at the base points, hence smooth everywhere by Bertini theorem.


Since the zero locus of every invariant or anti-invariant section is fixed by $(-1)\_A$, we can take as $C$ a general element of $|\mathcal{L}^2|^-$.


Note that the involution on $A$ induces a double cover $C \to C'$ ramified at $16$ points. Since $C^2=16$, by the genus formula it follows $g(C)=9$; thus the Riemann-Hurwitz formula implies $g(C')=1$.


**References.**


W. Barth: *Abelian surfaces with* $(1,\, 2)$-*polarization*, in Algebraic Geometry, Sendai, 1985, Adv.
Stud. Pure Math. **10**, North-Holland, Amsterdam, 1987, 41–84. 


See in particular Section 5.


