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We present an explicit construction of a compactification of the locus ofsmooth curves whose symmetric Weierstrass semigroup at a marked point is odd.The construction is an extension of Stoehr's techniques usingPinkham'sequivariant deformation of monomial curves by exploring syzygies. Asan application we prove the rationality of the locus for genus at most six.

On the locus of curves with an odd subcanonical marked point