Symmetric products of Gorenstein varieties (Q1184195)

For an affine algebraic variety \(X\) over a field \(k\), consider the action of the symmetric group \(S_ n\) on the cartesian product \(X^ n=X\times\dots\times X\) by permutation of the factors \((n\geq 2)\). The set of unordered \(n\)-uples \(X^ n/S_ n\) is an algebraic variety called the \(n\)-th symmetric product and denoted by \(X^{(n)}\). The descent of the Gorenstein property from \(X\) to \(X^{(n)}\) is studied. The main result reads that assuming \(X\) Gorenstein and \(k\) of characteristic 0 or of characteristic \(p>n\), then \(X^{(n)}\) is Gorenstein if and only if every connected component of \(X\) is either of even dimension, or is a smooth curve over \(k\).