A Note on Equimultiple Deformations

Abstract. While the tangent space to an equisingular family of curves can be discribed by the sections of a twisted ideal sheaf, this is no longer true if we only prescribe the multiplicity which a singular point should have. However, it is still possible to compute the dimension of the tangent space with the aid of the equimulitplicity ideal. In this note we consider families Lm = {(C, p) ∈ |L| × S | multp(C) = m} for some linear system |L| on a smooth projective surface S and a fixed positive integer m, and we compute the dimension of the tangent space to Lm at a point (C, p) depending on whether p is a unitangential singular point of C or not. We deduce that the expected dimension of Lm at (C, p) in any case is just dim |L| − m·(m+1) 2 + 2. The result is used in the study of triple-point defective surfaces in [ChM06a] and [ChM06b].