Introductory Examples of Blowups
in Rigid and Formal Geometry
Drew Moore
1
Intro
Blowups are helpful in the study of rigid spaces and formal schemes for the following reason:
if X is a formal scheme with rigid generic fiber X , and X̃ → X is an admissable blowup,
then the rigid generic fiber of X̃ = X remains the same. Here, admissable means blowing up
along an open ideal sheaf I. If X is a p-adic formal scheme, this just means that pn ∈ I(U )
for some n (possibly depending on the open set U ).
If X is algebraizable, i.e. comes from some scheme X, then the previous statement just
comes from the fact that blowups are isomorphisms away from the exceptional divisor.
In this note, I will first explain the blowup the formal disk at the origin, then I will collect
definitions related to blowups, then I will do a slightly more complicated blowup of the
formal disk.
2
Specialization of the Formal p-adic Disk
Let K be a p-adic, complete, non-archimedean field. Normalize the valuation so that
|p| = 1/p. Set OK = {x ∈ K : |x| ≤ 1} to be its ring of integers, m = {x ∈ K : |x| < 1} its
maximal ideal and k = OK /m its residue field.
Suppose X is a formal scheme over Spf OK . Denote by X its rigid generic fiber - it is a
rigid analytic variety over K. Denote by X its special fiber - it is an algebraic variety over
k.
There is a continuous map of topological spaces
sp : X → X.
Note this is not a map of ringed spaces, as the functions on X are characteristic p, while
those on X are p-adic. If x ∈ X , x extends uniquely to a Spf OK point Spf OK → X. The
1
specialization of this point gives a map Spec k → X. This is how the specialization map is
defined.
Let us explain this in detail when X = Spf OK hT i is the formal p-adic disk (and assume
for simplicity that K = K). Then X denote the rigid p-adic disk over K, while X is the
affine line A1/k over k. X is depicted in figure 1.
0
0
Figure 1: The formal disk Spf OK hT i. The special fiber is A1/k , while
the generic fiber is the disk {x ∈ K : |x| ≤ 1}. The white dot in the
special fiber represents an ∞ ‘removed.’
The points of X are the maximal ideals m of KhT i. Necessarily, the contraction of m to
K[T ] is also a maximal ideal, which is of the form (T −a) with a ∈ K (since we are assuming
K = K). Were a to have valuation strictly greater than 1, the formula
X
1
=−
a−n+1 T n
T −a
shows that T − a is a unit in KhT i (since |a|−n+1 → 0 as n → ∞). The maximal ideal
(T − a) thus gives a Spf OK point of X. It is the continuous map OK hT i → OK given by
T 7→ a. Its specialization is the k-point k[t] 7→ k with t 7→ ã ∈ k.
We see that if |a| < 1, sp(a) = 0 ∈ A1/k . Conversely, the “circle” |a| = 1 maps to the entire
open set A1/k \ {0} of the special fiber.
3
Blowing up the Formal p-adic Disk at the Origin of the
Special Fiber
We will blow up at the point 0 in the special fiber. This corresponds to the ideal (p, T ).
To blow up an ideal is to make that ideal locally principal. So we introduce a projective
2
coordinate u (with inverse v) such that when v 6= 0, we have up = T , and when u 6= 0,
p = vT .
We immediately see that the generic fiber is unchanged. Indeed since p is now invertible,
u = T /p, and v = p/T (when u is nonzero - i.e. when T is nonzero).
However, the special fiber has drastically changed. The equations become T = 0 (when
v 6= 0) and vT = 0 (when u 6= 0). We see that we are gluing a line with two axes along a
Gm , by which I mean we have an A1 and a P1 meeting transversely. Let X0 be the A1 , and
X1 the P1 .
The specialization map has changed. Figure 2 shows what has happened.
∞
10
0
2 = sp(2)
=
0
10)
0 glued
to ∞
sp(
2
2
Figure 2: The formal 5-adic disk blown up at the origin in the special
fiber. The blue line segment (X0 ) in the special fiber is an A1/k (the ∞
is removed), while the red (X1 ) is a P1/k . The red part (not including
the orange node) is where v 6= 0.
To understand this specialization map, you use the fact that the equations up = T and
p = vT give restrictions on the valuation on the values of T . Specifically, up = T (in the
region where v 6= 0) implies that |T | ≤ 1/p. While p = vT gives |T | ≥ 1/p (in the region
where u 6= 0.
4
Definitions
My main reference for this material is [2, Tag 01OF].
3
4.1
The Blowup of an Affine Scheme
Suppose X = Spec A, and Y ⊂ X is a closed subscheme cut out by the ideal J ⊂ A. Then
the blowup of X centered at Y is the proper X-scheme
M
J n = Proj(A ⊕ J ⊕ J 2 ⊕ · · · )
X̃ = Proj R(J)
R(J) :=
n≥0
The graded ring R(J) is called the Rees algebra of J. X̃ is characterized by the property
that the inverse image of Y under X̃ → X is a Cartier divisor, and is universal amongst
all such X-schemes. That is, if W → X is such that the pullback of Y to W is a Cartier
divisor, there exists a unique factorization W → X̃ → X.
4.2
Formal Blowups
We are interested in the role of blowups in formal and rigid geometry. So now suppose that
A is a topological ring with ideal of definition I. Set X = Spf A. The topological space
underlying X is the same as Spec A/I m for any m. The structure sheaf on Spf A is the limit
of the structure sheaves of Spec A/I m .
Suppose J ⊂ A is an open ideal, i.e. I N ⊂ J for some N . Then Y = Spf A/J is a closed
formal subscheme of X. We can form the formal proj of the Rees algebra of J, to get the
formal blowup of X at Y. Specifically, we have
X̃ = Projf R(J) := lim Proj R(J/I n )
←−
n>N
Remark: this is equivalent to completing X̃ = Proj R(J) with respect to the ideal sheaf
IOX̃ . This is the perspective that we will take when computing blowups. But the first
construction makes clearer the importance of taking an open ideal J - i.e. giving an admissable blowup.
4.3
Three Facts about Blowups
The following three facts are helpful for my own intuition about blowups.
Proposition. Suppose that J is an ideal on X = Spec A. Then the blowup of X centered
at J is isomorphic to the blowup of X centered at J n .
4
(Note: one does not need to restrict to affine X.) One can prove this in a couple ways.
First by the universal property, noting that an ideal sheaf J is invertible if and only if J n
is. Second, by noting that the d-uple map is an isomorphism. By that, I mean
M
M
Sdn
Proj S = Proj
Sn ∼
= Proj
Proposition. Suppose that X is a qcqs scheme, and that I, J are (finitely presented) ideal
sheaves on X. Let X 0 → X be the blowup centered at I, and X 00 → X 0 be the blowup
centered at J 0 = J OX 0 . Then X 00 → X is the blowup centered at IJ .
Proposition. If X is a qcqs scheme, and X 00 → X 0 → X is a sequence of blowups (centered
at ideals of finite presentation), then X 00 → X is a blowup.
5
A More Complicated Blowup of the Formal Disk
Consider the ideal J = (p3 , pT, T 2 ) = (p2 , T )(p, T ) ⊂ OK hT i. Would like to compute the
blowup X̃ of X̃ centered at J. Set
b3 = Projf OK hT i[W0 , W1 , W2 ]
P
X
where the grading is given by assigning 1 to each Wi , and the topology on the structure
sheaf is the p-adic topology.
b3 - that is, R(J) will be a quotient of OK hT i[W0 , W1 , W2 ].
X̃ will be a closed formal subscheme of P
X
The images of W0 , W1 , W2 in R(J) will correspond to the elements p3 , pT, T 2 in the first
graded piece of the Rees algebra.
i: R = QQ[x,y];
i: J = ideal(x^3, x*y, y^2);
i: reesIdeal(J)
2
2
o: = ideal (y*w - x*w , y*w - x w , x*w - w w )
1
2
0
1
1
0 2
o: Ideal of R[w , w , w ]
0
1
2
Figure 3: Macaulay 2 code to compute the blowup of J. Some output
has been omitted. We make the replacement x 7→ p and y 7→ T .
5
We use Macaulay2 to help compute the Rees algebra in figure 3. From it, we see
X̃ = Projf OK hT i[W0 , W1 , W2 ]/(pW2 − T W1 , p2 W1 − T W0 , pW12 − W0 W2 )
Let X̃ = X̃/k denote the special fiber of the blowup. Specifically, we have
X̃ = Proj k[T ][W0 , W1 , W2 ]/(T W1 , T W0 , W0 W2 )
We consider coordinates w0 = W0 /W2 and w1 = W1 /W2 (so W2 = 0 is the line at infinity).
Thus, we see that the fiber above T = 0 is the projective line w0 = 0 together with the line
at infinity. There is one point with T = a 6= 0, that is w0 = w1 = 0. Thus, the special fiber
looks like an affine line (coordinate T ) with two projective lines attached to each other, ∞
glued to 0.
X0 ∼
= A1/k
X1 ∼
= P1/k
X2 ∼
= P1/k
Figure 4: The special fiber X̃ of X̃. X0 is the T -line, X1 is where
T = W0 = 0, and X2 is where T = W2 = 0.
We now study the specialization map sp : X → X̃, where X is the rigid analytic disk.
First, suppose P is a Spf OK -point, whose specialization lands in X0 . So P = (t, [w0 : w1 :
1]) with t, w0 , w1 ∈ OK ,
p = tw1
p2 w1 = tw0
pw12 = w0
(†)
and |w0 |, |w1 | < 1. The first equation implies that |t| > 1/p. Conversely, if |t| > 1/p, the
above equations give w0 , w1 ∈ OK . Thus, 1/p < |t| ≤ 1 is the preimage of the T -line.
Now suppose that P is an Spf OK -point whose specialization lands in X1 , but not the node
X0 ∩ X1 . In particular, P = (t, [w0 : 1 : w2 ]) with t, w0 , w2 ∈ OK and |w0 | < 1, and
pw2 = t
p2 = tw0
6
p = w0 w2
Since |w2 | ≤ 1, the third equation ensures that |w0 | ≥ 1/p. This, the bound |w0 | < 1, and
the second equation imply that 1/p2 < |t| ≤ 1/p. Furthermore, the specialization of P isn’t
the node X1 ∩ X2 if and only if |t| = 1/p.
Finally, a similar sort of computation shows that the preimage of X2 is the region |t| ≤
1/p2 .
In figure 5, we’ve depicted the specialization map using a color key. Each region is described
as follows:
Xred = {|t| = 1}
Xorange = {1/p < |t| < 1}
Xyellow = {|t| = 1/p}
Xgreen = {1/p2 < |t| < 1/p}
Xblue = {|t| = 1/p2 }
Xpurple = {|t| < 1/p2 }
(†)
p2
0
−p
Figure 5: Depiction of the specialization map. On the right hand
side, is the special fiber. The preimage of each colored region is
depicted in the same color in the generic fiber on the left.
Let us make 2 remarks: First, had we chosen J = (π1 , T )(π2 , T ) with 0 < |π1 | < |π2 | <
1, the special fiber would be identical, but the specialization would be slightly different.
Namely, in (†), every 1/p would be replced by |π2 |, and every 1/p2 would be replaced by
|π1 |.
Second, the way we computed this blowup was probably inefficient. Specifically, since
J = (p, T )(p2 , T ) we could have first blown up centered at (p, T ), and then centered at the
7
pullback of (p2 , T ) (or the other way around). Doing it this way would allow us to just use
the computations we already made for the first example.
References
[1]
Siegfried Bosch and Werner Lütkebohmert. “Formal and rigid geometry. I. Rigid spaces”.
In: Math. Ann. 295.2 (1993), pp. 291–317. issn: 0025-5831.
[2] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu.
2016.
8