# first order deformation of maps and curves preserving dual graph

suppose that $$\mu:C \to X$$ be pointed stable map and $$G$$ be the dual graph of $$C$$.

Fulton and Pandharipande in their paper,FP_notes,define two linear spaces $$Def_G(\mu) \subset Def(\mu)$$ as first order deformation of pointed stable map preserving dual graph $$G$$ and $$Def_G(C) \subset Def(C)$$ first order deformation of the curve $$C$$ preserving dual $$G$$. My question is what do they mean by preserving dual graph?

• Wait, isn't this a duplicate of mathoverflow.net/questions/99298/… ? – Piotr Achinger Nov 17 '18 at 21:46
• Well I saw that post but i didnt get my answer and there just discussed about maps – Jigar Famil Nov 17 '18 at 21:48
• – JMP Dec 15 '18 at 4:04

## 1 Answer

My guess: a "deformation preserving dual graph" $$\tilde C$$ of $$C$$ over some Artinian local $$A$$ is a flat lifting $$\tilde C/A$$ which formally locally looks like $${\rm Spec}(A[x,y]/(xy))$$. Some would maybe call it an equisingular deformation. This is to be contrasted with "smoothing" deformations, which locally can look like $${\rm Spec}(A[x,y]/(xy-a)$$ for some $$a$$ in the maximal ideal of $$A$$. In simpler terms, this is just a deformation of the normalization $$C'$$ of $$C$$ together with a deformation of the divisor $$C' \times_C C'$$. If the components $$C_i$$ of $$C$$ are smooth (no self-intersections), this means deforming each $$C_i$$ separately and then deforming the intersection points $$p\in C_i\cap C_j$$ both as a point in $$C_i$$ and $$C_j$$. For example, if each $$C_i$$ has genus zero and hence no non-trivial deformations, a first order deformation of $$C$$ preserving the dual graph means choosing a tangent vector to each component at each singular point, modulo global vector fields on each $$C_i$$. Since $$H^0(\mathbf{P}^1, T)$$ is three-dimensional, this means that the dimension of such deformations equals $$\sum \max(0, n_i - 3)$$ where $$n_i$$ is the number of singular points on $$C_i$$ (self-intersections being counted twice), see the formula on p. 29.

In less concrete but more concise terms, such an "equisingular" curve is (probably) a flat and proper family over some base whose fibers have at worst nodal singularities and whose singular locus is also flat.

(I don't know about stable maps.)