We present an explicit method to produce upper bounds for the dimension of the moduli spaces of complete integral pointed Gorenstein curves with prescribed symmetric Weierstrass semigroups. We illustrate the method by handling explicitly a family of symmetric semigroups, providing an upper bound which is better than Deligne's bound.

An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

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

We provide a new lower bound for the dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the marked point, derived from the Deligne-Greuel formula and Pinkham's equivariant deformation theory. Using Buchweitz's description of the first cohomology module of the cotangent complex for monomial curves, we show that our lower bound improves a recently one given by Pflueger. By allowing semigroups running over suitable families of symmetric semigroups of multiplicity six, we show that this new lower bound is attained, and that the corresponding moduli spaces are non-empty and of pure dimension.

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