\beta \bigr) }$.

The main result of this paper is that the space ${\overline { {\mathcal M}} _{bir} \bigl( X, \beta \bigr) }$ is irreducible or empty, except in the case where $X$ has degree one and $\beta = - K_X$.

The idea of the proof is straightforward. First, prove that in the boundary of all the irreducible components of ${\overline { {\mathcal M}} _{bir} \bigl( X, \beta \bigr) }$ there are special morphisms of a given type (called in what follows “morphisms in standard form”). Second, show that the locus of morphisms in standard form is connected and contained in the smooth locus of ${\overline { {\mathcal M}} _{bir} \bigl( X, \beta \bigr) }$.

From these two facts we conclude immediately that the smooth locus of ${\overline { {\mathcal M}} _{bir} \bigl( X, \beta \bigr) }$ is connected. Since the smooth locus is dense, we deduce that ${\overline { {\mathcal M}} _{bir} \bigl( X, \beta \bigr) }$ is irreducible.

The methods used in the proof are of two different kinds. First, there are general techniques, mainly Mori’s Bend and Break Theorem, to break curves into components with low anticanonical degree. In the case where $X$ is the projective plane, 