  The two dimensional lattice Ginzburg-Landau hamiltonian is simulated
numerically for different values of the coherence length $\xi$ in units of the
lattice spacing $a$, a parameter which controls amplitude fluctuations. The
phase diagram on the plane $T-\xi$ is measured. Amplitude fluctuations change
dramatically the nature of the phase transition: for values of $\xi/a \simeq
1$, instead of the smooth Kosterlitz-Thouless transition there is a {\em first
order} transition with a discontinuity in the vortex density $v$ and a sharper
drop in the helicity modulus $\Gamma$. Both observables $v$ and $\Gamma$ are
analyzed in detail at the crossover region between first and second order which
occurs for intermediate values of $\xi/a$.
