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The Landau-Ginzburg-Wilson hamiltonian is studied for different values of theparameter $\lambda$ which multiplies the quartic term (it turns out that thisis equivalent to consider different values of the coherence length $\xi$ inunits of the lattice spacing $a$). It is observed that amplitude fluctuationscan change dramatically the nature of the phase transition: for small values of$\lambda$ ($\xi/a > 0.7$), instead of the smooth Kosterlitz-Thouless transitionthere is a {\em first order} transition with a discontinuous jump in the vortexdensity $v$ and a larger non-universal drop in the helicity modulus. Inparticular, for $\lambda$ sufficiently small ($\xi/a \cong 1$), the density ofbound pairs of vortex-antivortex below $T_c$ is so low that, $v$ drops to zeroalmost for all temperature $T<Tc$.Comment: 8 pages, 5 .eps figure

Phase transitions driven by vortices in two-dimensional superfluids and superconductors: From Kosterlitz-Thouless to first order