If $A$ is an abelian variety over a field, the Kummer variety $K_A$ associated to $A$ is obtained as the quotient of $A$ by the involution $\iota: a \mapsto -a$. If $A$ is a surface, it is well-known that a resolution of $K_A$ can be constructed simply by blowing up the image (under the quotient map) of the set of $2$-torsion points. Is this statement still true if $A$ has dimension at least $3$? Precisely, is the variety obtained from $K_A$ by blowing up the image of the locus of $2$-torsion points under the quotient map always smooth?
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Yes, it is smooth. What is different from dim 2, is that the resolution is not crepant, so the resolution of $K_A$ is not a Calabi--Yau variety.
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$\begingroup$ Thanks, Sasha. May I ask further: can you point me to a reference explaining why this variety is smooth? $\endgroup$ Commented Apr 12, 2013 at 23:12
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$\begingroup$ You can rewrite the construction as first blowing up all torsion point and then taking the quotient. In this case the fixed point locus of the involution is the union of exceptional divisors, so the quotient is smooth. Alternatively, you can look at the local model --- the blowup of ${\mathbb C}^n/\pm1$ and check it smoothness. $\endgroup$– SashaCommented Apr 13, 2013 at 3:02