Given an integral curve C of degree d in the projective space P^n over an algebraically closed field (for example, a plane conic), we discuss how one can give an upper bound on the dimension of the space of homogeneous polynomials F(x_0,...,x_n) of fixed degree, which are singular along C. This is achieved through a degeneration technique and is based on the upper semicontinuity theorem. This discussion will serve as motivation for working with general schemes and not just with classical varieties. Only minimal prior knowledge of algebraic geometry will be assumed. If time permits, we will discuss a second application of the upper semicontinuity theorem, which allows one to compare the dimension of an incidence correspondence in characteristic 0 and in positive characteristic.
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