SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber Gm and special fiber Ga.

What is such an example?


R[x,y]/(y^2-x^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 Gm, and the special fiber (setting \pi to zero) is the smooth part of a cuspdal cubic, yielding Ga.

  • $\begingroup$ In retrospect, I should have projectivized before removing the point, or maybe just started with a Proj. $\endgroup$ – S. Carnahan Mar 10 '10 at 4:39

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,(1-bt)-1]. Then Spec(S) is a group scheme using the multiplication rule

μ(t1,t2) = t1 + t2 - bt1t2.

and inverse

ν(t) = -t(1-bt)-1.

(Meaning, this is either a Hopf algebra structure or I view it as representing this group-valued functor on rings, depending on your theology.)

When b is invertible, then replacing t with the new coordinate s = 1-bt 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).


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