SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber G_{m} and special fiber G_{a}.
What is such an example?
SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber G_{m} and special fiber G_{a}. What is such an example? 


R[x,y]/(y^2x^3\pi x^2) gives the coordinate ring of a bad cubic curve, where \pi is a uniformizer in R. Remove the origin (which is the one singular point), and projectivize the curve by adding a point at infinity, so your group has an identity. The generic fiber (treating \pi as a unit) is then the smooth part of a nodal cubic, yielding G_{m}, and the special fiber (setting \pi to zero) is the smooth part of a cuspdal cubic, yielding G_{a}. 


I like Scott's elliptic curve construction, but here is another construction of essentially the same thing with more explicit formulas. Let your discrete valuation ring be R = k[[b]] and your group scheme be Spec of the ring S = R[t,(1bt)^{1}]. Then Spec(S) is a group scheme using the multiplication rule μ(t_{1},t_{2}) = t_{1} + t_{2}  bt_{1}t_{2}. and inverse ν(t) = t(1bt)^{1}. (Meaning, this is either a Hopf algebra structure or I view it as representing this groupvalued functor on rings, depending on your theology.) When b is invertible, then replacing t with the new coordinate s = 1bt makes this isomorphic to the multiplicative group scheme (and, in fact, that's how the above formulas for multiplication and inversion are easiest to obtain). When b is zero, this reduces to the additive group. 


One more point of view : if X is a 2x2 matrix, its centralizer in PGL(2) is  for generic X  isomorphic to Gm; however, it is isomorphic to Ga if X is nonzero and nilpotent. This gives a typical naturally occurring family (you can, of course, restrict it to get a family over a dvr). 

