By Dan Abramovich, Angelo Vistoli (auth.), Geir Ellingsrud, William Fulton, Angelo Vistoli (eds.)

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**Example text**

Let L be a universal deformation space of Xo; r acts on L. Let V' be a small etale neighborhood of 0 in A 1 , and let r act on V', such that a generator sends t to -to Choose a non-constant r-equivariant map V' -+ L; this yields a family Y' of stable curves on V' whose fiber over 0 is exactly Xo; r also acts on Y. Let V be the union of two copies of V' glued at 0; there are two families Y1 and Y2 on V obtained by attaching two copies of Y' at X o, one using the identity, the other using f. These will not be isomorphic in general, not after an automorphism of V, and not even after going to an etale neighborhood of 0 in V.

Also, if m > 0, (U~/z,) [m] maps onto the m-th graded quotient of the latter filtration since it surjects to S(gyNy\y,xz,z, Therefore by (27) the map --+ U p' 1----+ U must also be surjective. Thus the only thing left to check is that the natural surjection Z/Z') ( m [O]V y / y , [m] _ SOyNy\y,xz'Z ( Z/Z') [O]V y / y , [m-l] is actually injective. But this is immediately seen by writing this map in local coordinates. 2. i) v~~f, is theU-submoduleof'D'i/z' (n~/z"WY/y,) generated by p. The natural filtration {U[m]} on U induces an increasing filtration on 'D yz/z' / y , such that / y , = OY'· P Gro ( 'D yZ/Z') and, ifm > 0, Grm ( 'DyZ/Z') /y, is itself filtered with associated gradeds being given by ii) Let Yo denote a fiber of py" Suppose we are given a projection Z~Yo locally around a point y E Yo (where, abusing notation, we temporarily denote the neighborhood of y E Z again by the letter Z).

It is easy to see that the property of a morphism f : X -+ M being stable is independent of the choice of the projective embedding of M. Note that, given a stable map f : X -+ M, one has a well-defined triple of rational numbers: One can define a functor of families of stable maps with fixed invariants A, B and C. This is somewhat subtle, since the sheaf wx(D) is not invertible, and saturation does not commute with base change. This can be resolved either by restricting to "allowable" deformations on which the saturation (wx(D))[m) does commute with base change (this is discussed in an unpublished work by Kollar), or by endowing the surface X with the structure of a Deligne-Mumford stack using the log-Gorenstein covers (this has not been carried out in the literature).