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consider
the example: f = r2 + x2, with the critical point (0; 0). Then change f to f = r2 + x2 + 10(cid:0)30,
which causes the critical point to cease to exist. This example illustrates the fact that:
Isolated critical points2 are structurally unstable, thus not (generally) interesting.
Second: imagine now that f depends on some extra parameter f = f (x; r; h), such that the assump-
tions in (1.2) apply for (x; r; h) = (0; 0; 0) | here h small and nonzero produces an \arbitrary"
(smooth) perturbation to the dynamical system in (1.1). Consider now the system of equations:
f (x; r; h) = 0;
and
fx(x; r; h) = 0:
(2.4)
2These are points such that there is a neighborhood in (x; r) space where there is no other critical point.
6
6
Rosales Bifurcations: baby normal forms.
Now (0; 0; 0) is a solution to this system, and the Jacobian matrix
J = 0
B
@
f
x(0; 0; 0)
fr(0; 0; 0)
1
fxx(0; 0; 0) fxr(0; 0; 0) C
0
= 0
1
2 fxr(0; 0; 0)
A @
B (cid:0)
5
(2.5)
1
C
A
is non-singular there. Thus the implicit function theorem guarantees that there is a (unique) smooth
curve of solutions x = x(h) and r = r(h) to (2.4), with x(0) = 0 and r(0) = 0. Along this curve, for
h small enough, it is clear that: fr(x; r; h) = 0 and fxx(x; r; h) = 0. Thus, modulo normalization,
(2.3) is valid along the curve | for h small enough. This (cid:12)nishes the proof. | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
Thus, modulo normalization,
(2.3) is valid along the curve | for h small enough. This (cid:12)nishes the proof.
Remark 5 In the proof of structural stability in the prior remark, we assumed that the pertur-
bations to the dynamical system in (1.1) had the form
dx
dt
= f (x; r; h);
(2.6)
with the dependence in the \extra" parameter h being smooth. This sounds reasonable, but (clearly)
it does not cover all possible (imaginable or non-imaginable) perturbations. For example, we could
consider \perturbations" of the form
dx
dt
= f (x; r) + h
d2x
dt2 :
(2.7)
What \small" means in this case is not easy to state (and we will not even try here). However, this
example should make it clear that: when talking about structural stability, for the concept to
even make sense, the dynamical system must be thought as belonging to some \class"
| within which the idea of \close" makes sense. Further, the answer to the question: is this
system structurally stable? will be a function of the class considered.
Let us now get back to the system in (1.1), with the assumptions in (2.3), and let us
study the bifurcation that occurs in this case: the Saddle Node bifurcation.
We proceed formally (cid:12)rst, by expanding in Taylor series and writing the equation in the form
dx
dt
= r
(cid:0)
x2 + O(r2; rx; x3);
(2.8)
where all the information in (2.3) has been used. We now look at this equation in a small (rectan-
gular) neighborhood of the origin, characterized by
< (cid:15)
and
x
j
j
< (cid:15)2;
r
j
j
(2.9)
6
6
Rosales Bifurcations: baby normal forms.
6
where 0 < (cid:15)
(cid:28)
1. Then the (cid:12)rst two terms on the | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
6
where 0 < (cid:15)
(cid:28)
1. Then the (cid:12)rst two terms on the right in (2.8) are O((cid:15)2), while the rest is O((cid:15)3). We
thus argue that the behavior of the system in the neighborhood given by (2.9) is well approximated
by the equation
dx
dt
x2:
= r
(cid:0)
(2.10)
This is the Normal form for a Saddle Node bifurcation | see Strogatz book for
a description of its behavior.
Remark 6 A natural question here is: Why the scaling in (2.9)? Such a question can
only be answered \after the fact", with the answer being (basically) \because it works". Namely,
after we have (cid:12)gured out what is going on, we can explain why the scaling in (2.9) is the right one
to do. As follows: at a Saddle Node bifurcation | say, at (x; r) = (0; 0) | a branch of critical
point solutions | say x = X1(r) | turns \back" on itself.3 Thus, on one side of the value r = 0,
no critical point exist, while on the other side two are found, say at: x = X1(r) and x = X2(r).
Locally, these two curves can be joined into a single one by writing r = R(x). Then r = R(x) has
either a maximum (or a minimum) at x = 0. Hence it can, locally, be approximated by a parabola.
Hence the scaling in (2.9) is the right one. Any other scaling would miss the fact that we have a
branch of critical points turning around.
Those with a mathematical mind will probably not be very satis(cid:12)ed with this explanation. For them,
the theorem below might do the trick. However, note that this theorem is just a proof that equation
(2.10) is the right answer, showing that (2.9) works. It does not give any reason (or method) that
would justify (2 | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
0) is the right answer, showing that (2.9) works. It does not give any reason (or method) that
would justify (2.9) \a priori". Simply put: advance in science and mathematics requires places at
which \insight" is needed, and (2.9) is an example of this; perhaps a very simple example, but one
nonetheless.
Theorem 1 With the hypothesis in equation (2.9), there exists a neighborhood of the origin, and
there a smooth coordinate transformation (x; t)
(X; T ) of the form
!
X = x (cid:8)(x)
and
dT
dt
= (cid:9)(x; r);
(2.11)
such that (1.1) is transformed into
dX
dT
= r
(cid:0)
X 2 | that is, the normal form in equation (2.10).
Furthermore: (cid:8)(0) = 1 and (cid:9)(0; 0) = 1 | thus: X
x and T
(cid:25)
(cid:25)
t close to the origin.
3Note that this is the reason that this type of bifurcation is also known by the name of turning point bifurcation.
Rosales Bifurcations: baby normal forms.
7
IMPORTANT: the de(cid:12)nition for the transformed time is meant to be done along the solutions.
That is: in the equation
dT
dt
= (cid:9)(x; r), x = x(t) is a solution of equation (1.1).
Proof: Using the implicit function theorem, we see that f (x; r) = 0 has a unique (also smooth)
solution r = R(x) in a neighborhood of the origin: f (x; R(x))
0; which satis(cid:12)es R(0) = 0. It
(cid:17)
is easy to see that (dR=dx)(0) = 0 and (d2R=dx2)(0) = 2 also apply. Thus R = x2 (cid:8)(x)2; where
(cid:8) is smooth and (cid:8)(0) = | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
Thus R = x2 (cid:8)(x)2; where
(cid:8) is smooth and (cid:8)(0) = 1 | this is the (cid:8) which appears in equation (2.11).
Because f (x; R(x))
(cid:17)
0, we can write f (x; r) = (cid:11)(x; r) (r
R(x)) = (cid:11)(x; r) (r
(cid:0)
(cid:0)
X 2); where (cid:11) is
smooth and does not vanish near the origin | in fact: (cid:11)(0; 0) = 1:
De(cid:12)ne now (cid:9) = ((cid:8) + x(cid:8)
0
) (cid:11); where the prime indicates di(cid:11)erentiation with respect to x. It is
then easy to check that (cid:9)(0; 0) = 1, and that with this de(cid:12)nition (2.11) yields
dX
dT
X 2.
= r
(cid:0)
QED.
Problem 1 Implement a reduction to normal form, along lines similar to those used in theo-
rem 1, by formally expanding the coordinate transformation up to O((cid:15)2) | where (cid:15) is as in equation
(2.9). To do so, write the dynamical system in the expanded form
dx
dt
= r
x2 + a r x + b x3 + O((cid:15)4);
(cid:0)
(cid:15)3)
(cid:15)2)
O(
O(
{z
}
{z
|
}
|
where a and b are constants. Then expand the transformation
x = X + (cid:11) X 2 + O((cid:15)3);
dt
dT
= 1 + (cid:12) X + O((cid:15)2);
and (cid:12)nd what values the coe(cid:14)cients (cid:11) and (cid:12 | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
);
and (cid:12)nd what values the coe(cid:14)cients (cid:11) and (cid:12) must take so that
dX
dT
= r
(cid:0)
X 2 + O((cid:15)4):
(2.12)
(2.13)
(2.14)
(2.15)
This process can be continued so as to make the error term in (2.15) as high an order in (cid:15) as
desired | provided that f in (1.1) has enough derivatives. We point out here that: theorem 1
requires f to have only second order continuous derivatives to apply. By
contrast, the process here requires progressively higher derivatives to exist | it, however, has the
advantage of giving explicit formulas for the transformation.
Rosales Bifurcations: baby normal forms.
8
3 Transcritical bifurcations.
We now go back to the considerations in the introduction (section 1), and add one extra hypothesis
at the bifurcation point (x0; r0) = (0; 0). Namely: we assume that there is a smooth
branch x = (cid:31)(r) of critical points that goes through the bifurcation point.
Taking successive derivatives of the identity f ((cid:31)(r); r)
0, and evaluating them at r = 0 (where
(cid:17)
(cid:31) = f = fx = 0), we obtain:
f
r(0; 0) = 0 and frr(0; 0) =
2
(cid:22) fxx(0; 0)
(cid:0)
(cid:0)
2 (cid:22) fxr(0; 0);
(3.16)
(0). As before, we assume that the coe(cid:14)cients for which we have no information are
where (cid:22) =
d(cid:31)
dr
non-zero, and normalize (by scaling r and x in equation (1.1), if needed) so that: fxx(0; 0) =
2
(cid:0)
and frx(0; | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
1.1), if needed) so that: fxx(0; 0) =
2
(cid:0)
and frx(0; 0) = 1. Thus, at (x; r) = (0; 0), we have:
f = fx = fr = 0;
fxx =
2;
(cid:0)
fxr = 1;
and frr = 2 a;
(3.17)
where a is a constant. In fact, (3.16) and (3.17), show that a = (cid:22)2
(cid:22) = ((cid:22)
(cid:0)
1=2)2
(cid:0)
(cid:0)
1=4. Thus
0. We do not know what the exact value of (cid:22) is, however, as usual (for
1 + 4 a = (2 (cid:22)
1)2
(cid:0)
(cid:21)
generality) we exclude the equal sign in this last inequality as \too special". Thus:
Assume 1 + 4 a > 0:
(3.18)
As in the case of the Saddle Node bifurcation, the next step is to use (3.17) to expand the equation
(1.1). This yields:
dx
dt
= r x
(cid:0)
x2 + a r2 + O(x3; r x2; r2 x; r3):
(3.19)
We now assume4 that both r and x are small, of size O((cid:15)), where 0 < (cid:15)
keeping up to terms of O((cid:15)2) on the right (leading order) in (3.19), we obtain the equation:
(cid:28)
1. Then,
dx
dt
= r x
(cid:0)
x2 + a r2 = (x
(cid:0) (cid:0)
(cid:27) r) (x
1
(cid:27) r);
2
(cid:0)
where (cid:27)1 =
1
2
1 + p1 + 4 a
(cid:16)
(cid: | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
:0)
where (cid:27)1 =
1
2
1 + p1 + 4 a
(cid:16)
(cid:17)
and (cid:27)2 =
1
2
1 p
(cid:0)
(cid:16)
1 + 4 a
(cid:17)
and R = p1 + 4 a r, this last equation takes the form:
. In terms of the variables X = x
dX
dt
= R X
X 2;
(cid:0)
which is the Normal form for a Transcritical bifurcation.
4Compare this with (2.9).
(3.20)
(cid:27)2 r
(cid:0)
(3.21)
Rosales Bifurcations: baby normal forms.
9
Remark 7 The hypothesis 1 + 4 a > 0 in (3.18) is very important. For, write equation (3.19) in
the form:
dx
dt
=
x
(cid:0)
(cid:0)
(cid:18)
2
r +
(cid:19)
1
2
1 + 4 a
4
r2 + O(x3; r x2; r2 x; r3):
(3.22)
Then, if 1 + 4 a < 0, the leading order terms on the right in this equation would be a negative de(cid:12)nite
quadratic form. This would imply that (x; r) = (0; 0) is the only critical point in a neighborhood of
the origin | i.e. that (x; r) = (0; 0) is an isolated critical point. As explained in remark 4, such
points are (generally) of little interest.
On the other hand, 1 + 4 a = 0 would lead to a double root of the right hand side in (3.19) (at
leading order). In principle this can be interpreted as a \limit case" of the transcritical bifurcation,
where the two branches of critical points that cross at the origin, become tangent there. However:
this is an extremely structurally unstable situation, where the local details of what actually happens
are controlled by high | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
at the origin, become tangent there. However:
this is an extremely structurally unstable situation, where the local details of what actually happens
are controlled by high order terms | hence, again, this is a situation of little (general) interest.
Theorem 2 If the function f = f (x; r) is su(cid:14)ciently smooth, the assumptions in equations
(3.17) and (3.18) guarantee that the f = f (x; r) = 0 has (exactly) two branches of solutions in
a neighborhood of the origin. Furthermore, let this branches be given by x = (cid:31)1(r) and x = (cid:31)2(r).
Then (cid:31)1(r) = (cid:27)1 r + O(r2) and (cid:31)2(r) = (cid:27)2 r + O(r2).
Sketch of the proof: The calculations leading to equations (3.19) and (3.20) show that:
f (x; r) =
(x
(cid:0)
(cid:0)
(cid:27)1 r) (x
(cid:0)
2 r) + O(x ; r x2; r2 x; r3):
3
(cid:27)
(3.23)
Let x = r X. Then
f (x; r) =
r2 (X
(cid:0)
(cid:0)
(cid:27)1) (X
(cid:0)
(cid:27)2) + r3 O(X 3; X 2; X):
Thus, g = g(X; r) =
f
(cid:0) r2
satis(cid:12)es:
g(X; r) = (X
(cid:27)1) (X
(cid:0)
(cid:0)
(cid:27)2) + r h(X; r);
where h is some non-singular function. We note now that:
g((cid:27)p; 0) = 0 and gX((cid:27)p; 0) = ((cid:27)p
(cid:27)q) = 0;
(cid:0)
( | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
:27)p; 0) = ((cid:27)p
(cid:27)q) = 0;
(cid:0)
(3.24)
(3.25)
(3.26)
6
Rosales Bifurcations: baby normal forms.
10
where
p; q
=
1; 2
. Then the implicit function theorem guarantees that there exist smooth
g
f
g
f
solutions X = Xn(r) to the equations:
g(Xn; r) = 0 and Xn(0) = (cid:27)n | where n = 1 or n = 2.
Then (cid:31)n = r Xn | for n = 1; 2 | are the two functions in the theorem statement.
Why are there no other solutions? Well, once we have (cid:31)1 and (cid:31)2, we can write f = (x
where = (x; r) does not vanish at the origin | in fact: (0; 0) = fxx(0; 0) =
(cid:0)
(cid:31)1)(x
(cid:31)2) ,
(cid:0)
(cid:0)
2.
QED.
The arguments made to obtain equations (3.17) and (3.18) depend on the existence of the smooth
branch of critical points x = (cid:31)(r). But the existence of this branch is not then used in the arguments
leading to the normal form (3.21). We explicitly exploit this existence in what follows below, and
use it to get a better handle on transcritical bifurcations. Thus, without loss of generality:5
Assume that (cid:31)
0.
(cid:17)
(3.27)
Then we can write f = x G(x; r); where G(0; 0) = 0 | since fx(0; 0) = 0. Other than this, we
assume that G is \generic", so that its derivatives do not vanish. In particular, we normalize the
(cid:12)rst order derivatives so that Gr(0; 0) =
(cid: | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
derivatives do not vanish. In particular, we normalize the
(cid:12)rst order derivatives so that Gr(0; 0) =
(cid:0)
Gx(0; 0) = 1 | this normalization is consistent with
the one used in (3.17), where we must take a = 0.
At this point we can invoke the implicit function theorem, that tells us that there is a function
x = z(r) such that G(z; r) = 0, with z(0) = 0 and dz=dr(0) = 1 | note that, in this case (cid:27)1 = 1
and (cid:27)2 = 0. Again, we use this function to factor G in the form G = (x
z(r)) H(x; r); where
(cid:0)
It follows then that we can write equation (1.1) in the form:
H(0; 0) =
1:
(cid:0)
Thus, if we introduce a new time T by dT =dt =
H;
and change parameter6 r
the equation is transformed into its Normal Form:
1 dx
H dt
=
(cid:0)
z(r) x + x2:
(cid:0)
dx
dT
= R x
x2:
(cid:0)
(3.28)
R = z(r);
!
(3.29)
The above is, clearly, the equivalent of theorem 1 for transcritical bifurcations:
a proof of the existence of a local transformation into normal form.
5If needed, the change of variables x
6Note that, for r and x small, R
x
!
(cid:0)
r and T
(cid:25)
(cid:31) will do the trick.
t. Thus both R and T are acceptable new variables.
(cid:25)
Rosales Bifurcations: baby normal forms.
11
Remark 8 Note that, because G above is \generic", the situation is structurally sta-
ble. However, this depends on the assumption that there is a branch of solutions. Transcritical
bifurcations are not structurally stable without an assumption of this type.
Problem 2 Assume that equation (3.27), and the | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
bifurcations are not structurally stable without an assumption of this type.
Problem 2 Assume that equation (3.27), and the normalizations immediately below it, apply.
Then, for x and r both small and O((cid:15)) | where 0 < (cid:15)
1 | implement a reduction to normal
(cid:28)
form, by formally expanding the coordinate transformation up to two orders in (cid:15). To do so, write
the dynamical system in the expanded form
dx
dt
= r x
0
x2 + b x3 + b r x2 + b r2 x + O((cid:15)4);
(cid:15)3)
O(
{z
}
|
}
1
2
(cid:0)
(cid:15)2)
O(
{z
|
where b0, b1, and b2 are constants. Then expand the transformation
dt
dT
= 1 + (cid:12)0 x + (cid:12)1 R + O((cid:15)2);
r = R + (cid:13) R2 + O((cid:15)3);
and (cid:12)nd what values the coe(cid:14)cients (cid:12)0, (cid:12)1, and (cid:13) must take so that
dx
dT
= R x
(cid:0)
x2 + O((cid:15)4):
(3.30)
(3.31)
(3.32)
(3.33)
This process can be continued so as to make the error term in (3.33) of arbitrarily high order in (cid:15) |
provided that f in (1.1) has enough derivatives. We point out here that: the derivation lead-
ing to equation (3.29) requires f to have only second order continuous
derivatives to apply. By contrast, the process here requires progressively higher derivatives
to exist | it, however, has the advantage of giving explicit formulas for the transformation.
Problem 3 In problem 3.2.6 in the book by Strogatz, a process somewhat analogous to the one
in | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
for the transformation.
Problem 3 In problem 3.2.6 in the book by Strogatz, a process somewhat analogous to the one
in problem 2 is introduced. Basically, Strogatz tells you to do the following:
Consider the system
dx
dt
= R x
(cid:0)
x2 + a x3 + O(x4);
where R = 0 and a are constants. Introduce now a transformation (expanded) of the form
x = X + b X 3 + O(X 4);
(3.34)
(3.35)
6
Rosales Bifurcations: baby normal forms.
12
where b is a constant. Then show that b can be selected so that the equation for X has the form
dX
dt
= R X
(cid:0)
X 2 + O(X 4):
(3.36)
Thus the third order power is removed. The process can be generalized to remove arbitrarily high
powers of X from the equation.
Question: This process is simpler than the one employed in problem 2:
it involves neither trans-
forming the independent variable t, nor the parameter R. Why is it not appropriate for
reducing an equation to normal form near a transcritical bifurcation?
4 Pitchfork bifurcations.
We now go back to the considerations in the introduction (section 1), and add two extra hypotheses
at the bifurcation point (x0; r0) = (0; 0), one of them being the same one that was introduced in
section 3 for the transcritical bifurcations. Namely, we assume that:
A. There is a smooth branch x = (cid:31)(r) of critical points that goes through
the bifurcation point (0; 0).
B. The problem has right-left symmetry across the branch of critical
points x = (cid:31)(r). Speci(cid:12)cally, there is smooth bijection x
valid in a neighborhood of the branch x = (cid:31); such that:
X = X(x; r);
!
| Equation (1.1) is invariant under the transformation: X = f (X; r).
_
| (cid:31) is a (cid:1 | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
1.1) is invariant under the transformation: X = f (X; r).
_
| (cid:31) is a (cid:12)xed curve for the transformation: X((cid:31)(r); r) = (cid:31)(r).
| x < (cid:31)(r) =
)
X > (cid:31)(r)
and
x > (cid:31)(r) =
)
X < (cid:31)(r).
Without any real loss of generality, assume that f (x; r) is an odd function
of x. Then (cid:31) = 0, X =
x, and (1.1) becomes:
(cid:0)
dx
dt
= x g((cid:16); r); where
(cid:16) = x2:
(4.37)
Rosales Bifurcations: baby normal forms.
13
The bifurcation condition (1.2) yields g(0; 0) = 0. Other than this, we assume that
g is generic. After appropriate re-scaling of the variables, we thus have
g(0; 0) = 0;
gr(0; 0) = 1;
and g(cid:16)(0; 0) = (cid:23) =
1:
(cid:6)
(4.38)
Note that the sign of g(cid:16)(0; 0) cannot be changed by scalings!
Problem 4 Expand g in (4.37) in powers of (cid:16) and r. Show that, in a small neighborhood of the
origin (of appropriate shape | see (2.9) and (3.19 { 3.20)), the leading order terms in the equation
reduce to the normal form for a pitchfork bifurcation:
dx
dt
= r x + (cid:23) x3:
Problem 5 In a manner analogous to the ones in problems 1 (saddle-node bifurcations) and
2 (transcritical bifurcations) introduce new variables (via formal expansions) x
R, that reduce equation (4.37) to normal form:
r
!
d | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
cations) introduce new variables (via formal expansions) x
R, that reduce equation (4.37) to normal form:
r
!
dX
dT
= R X + (cid:23) X 3:
X, t
!
!
T , and
HINT: First
(cid:15)
Expand g in a Taylor series g = r + (cid:23) (cid:16) + a2 r2 + a1 r (cid:16) + a0 (cid:16) 2 + : : :
and substitute
this expansion into the equation. Second
(cid:15)
Assume an appropriate size scaling for the variables x
and r in terms of a small parameter 0 < (cid:15)
1. This scaling should be consistent with the normal
form for the equation.7 It is very important since it assures that the ordering in the expansions is kept
(cid:28)
straight, without higher order terms being mixed with lower order ones. Third
Introduce expansions
for R = R(r) = r + o(r) and dT =dt = H(x2; r) = 1 + o(1). IMPORTANT: Notice that, because of
(cid:15)
the symmetry in the equation, it must be that x = X and the expansion for dT =dt must involve even
powers of x only. Fourth
(cid:15)
Substitute the expansions in the equation, and select the coe(cid:14)cients to
eliminate the higher orders beyond the normal form. Carry this computation to ONE ORDER
ONLY: What are the dominant terms in the expansions, beyond R
r and dT =dt
1.
(cid:24)
(cid:24)
Problem 6 Prove that a transformation with the properties stated in problem 5 actually exists |
this in a way similar to the one used in theorem 1 for saddle-node bifurcations, and above equation
(3.29) for transcritical bifurcations.
HINT: Show that g((cid:16); r) = 0 has a solution of the form (cid:16) =
(cid:23) R(r). Use this solution to \factor" g
(cid:0)
as a product, and | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
cid:16) =
(cid:23) R(r). Use this solution to \factor" g
(cid:0)
as a product, and substitute the result into the equation. It should then be obvious how to proceed.
7It is the same scaling required by problem 4.
Rosales Bifurcations: baby normal forms.
14
5 Problem Answers.
The problem answers will be handed out with the answers to the problem
sets.
MIT OpenCourseWare
http://ocw.mit.edu
18.385J / 2.036J Nonlinear Dynamics and Chaos
Fall 2014
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-385j-nonlinear-dynamics-and-chaos-fall-2014/068224e607b5dcde1629732a987ec9f2_MIT18_385JF14_BabyNormlFms.pdf |
THE MODULI SPACE OF CURVES
1. The moduli space of curves and a few remarks about its
construction
The theory of smooth algebraic curves lies at the intersection of many branches
of mathematics. A smooth complex curve may be considered as a Riemann sur
face. When the genus of the curve is at least 2, then it may also be considered as a
hyperbolic two manifold, that is a surface with a metric of constant negative curva
ture. Each of these points of view enhance our understanding of the classification
of smooth complex curves. While we will begin with an algebraic treatment of the
problem, we will later use insights offered by these other perspectives.
As a first approximation we would like to understand the functor
Mg : {Schemes} � {sets}
that assigns to a scheme Z the set of families (up to isomorphism) X � Z flat over
Z whose geometric fibers are smooth curves of genus g.
There are two problems with this functor. First, there does not exist a scheme
that represents this functor. Recall that given a contravariant functor F from
schemes over S to sets, we say that a scheme X(F ) over S and an element U (F ) ⊗
F (X(F )) represents the functor finely if for every S scheme Y the map
given by g � g�U (F ) is an isomorphism.
HomS (Y, X(F )) � F (Y )
Example 1.1. The main obstruction to the representability (in particular, to the
existence of a universal family) of Mg is curves with automorphisms. For instance,
fix a hyperelliptic curve C of genus g. Let π denote the hyperelliptic involution of
C. Let S be a K3-surface with a fixed point free involution | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
olution of
C. Let S be a K3-surface with a fixed point free involution i such that S/i is an
Enriques surface E. To be very concrete let C be the normalization of the plane
curve defined by the equation y2 = p(x) where p(x) is a polynomial of degree 2g + 2
with no repeated roots. The hyperelliptic involution is given by (x, y) ⊂� (x, −y).
Let Q1, Q2, Q3 be three general ternary quadratic forms. Let the K3-surface S be
defined by the vanishing of the three polynomials Qi(x0, x1, x2) + Qi(x3, x4, x5) = 0
with the involution that exchanges the triple (x0, x1, x2) with (x3, x4, x5). Consider
the quotient of C × S by the fixed-point free involution π × i. The quotient is a
non-trivial family over the Enriques surface E; however, every fiber is isomorphic to
C. If Mg were finely represented by a scheme, then this family would correspond
to a morphism from E to it. However, this morphism would have to be constant
since the moduli of the fibers is constant. The trivial family would also give rise to
the constant family. Hence, Mg cannot be finely represented.
There are two ways to remedy this problem. The first way is to ask a scheme to
only coarsely represent the functor. Recall the following definition:
1
Definition 1.2. Given a contravariant functor F from schemes over S to sets,
we say that a | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
a contravariant functor F from schemes over S to sets,
we say that a scheme X(F ) over S coarsely represents the functor F if there is a
natural transformation of functors � : F � HomS (�, X(F )) such that
(1) �(spec(k)) : F (spec(k)) � HomS (spec(k), X(F )) is a bijection for every
algebraically closed field k,
(2) For any S-scheme Y and any natural transformation � : F � HomS (�, Y ),
there is a unique natural transformation
Φ : HomS (�, X(F )) � HomS (�, Y )
such that � = Φ ∩ �.
The main theorem of moduli theory asserts that there exists a quasi-projective
moduli scheme coarsely representing the functor Mg .
Alternatively, we can ask for a Deligne-Mumford stack that parameterizes smooth
curves. Below we will give a few details explaining how both constructions work.
There is another serious problem with the functor Mg . Most families of curves
in projective space specialize to singular curves. This makes it seem unlikely that
any moduli space of smooth curves will be proper. This, of course, is in no way
conclusive. It is useful to keep the following cautionary tale in mind.
Example 1.3. Consider a general pencil of smooth quartic plane curves specializing
to a double conic. To be explicit fix a general, smooth quartic F in P2 . Let Q be
a general conic. Consider the family of curves in P2 given by
Ct : Q2 + tF.
I claim that after a base change of order 2, the central fiber of this family may be
replaced by a smooth, hyperelliptic curve of genus 3. The total space of this | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
central fiber of this family may be
replaced by a smooth, hyperelliptic curve of genus 3. The total space of this family
is singular at the 8 points of intersection of Q and F . These are ordinary double
points of the surface. We can resolve these singularities by blowing up these points.
Figure 1. Quartics specializing to a double conic.
We now make a base change of order 2. This is obtained by taking a double cover
branched at the exceptional curves E1, . . . , E8. The inverse image of the proper
transform of C0 is a double cover of P1 branched at the 8 points. In particular,
2
it is a hyperelliptic curve of genus 3. The inverse image of each exceptional curve
is rational curve with self-intersection −1. These can be blown-down. Thus, after
base change, we obtain a family of genus 3 curves where every fiber is smooth.
Exercise 1.4. Consider a general pencil of quartic curves in the plane specializing
to a quartic with a single node. Show that it is not possible to find a flat family
of curves (even after base change) that replaces the central fiber with a smooth
curve. (Hint: After blowing up the base points of the pencil, we can assume that
the total space of the family is smooth and the surface is relatively minimal. First,
assume we can replace the central fiber by a smooth curve without a base change.
Use Zariski’s main theorem to show that this is impossible. Then analyze what
happens when we perform a base change.)
The previous exercise shows that the coarse | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
Then analyze what
happens when we perform a base change.)
The previous exercise shows that the coarse moduli scheme of smooth curves
(assuming it exists) cannot be proper. Given that curves in projective space can
become arbitrarily singular, it is an amazing fact that the moduli space of curves
can be compactified by allowing curves that have only nodes as singularities.
Definition 1.5. Consider the tuples (C, p1, . . . , pn) where C is a connected at
worst nodal curve of arithmetic genus g and p1, . . . , pn are distinct smooth points
of C. We call the tuple (C, p1, . . . , pn) stable if in the normalization of the curve
any rational component has at least three distinguished points—inverse images of
nodes or of pi—and any component of genus one has at least one distinguished
point.
Note that for there to be any stable curves the inequality 2g − 2 + n > 0 needs
to be satisfied.
Definition 1.6. Let S be a scheme. A stable curve over S is a proper, flat family
C � S whose geometric fibers are stable curves.
Theorem 1.7 (Deligne-Mumford-Knudsen). There exists a coarse moduli space
Mg,n of stable n-pointed, genus g curves. Mg,n is a projective variety and contains
the coarse moduli space Mg,n of smooth n-pointed genus g curves as a Zariski open
subset.
One way to construct the coarse moduli scheme of stable curves is to consider
pluri-canonically embedded curves, that is | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
moduli scheme of stable curves is to consider
pluri-canonically embedded curves, that is curves embedded in projective space
P(2n−1)(g−1)−1 by their complete linear system |nKC | for n → 3. A locally closed
subscheme K of the Hilbert scheme parameterizes the locus of n-canonical curves
of genus g. The group P GL((2n − 1)(g − 1)) acts on K. The coarse moduli scheme
may be constructed as the G.I.T. quotient of K under this action. The proof that
this construction works is lengthy. Below we will briefly explain some of the main
ingredients. We begin by recalling the key features of the construction of the Hilbert
scheme. We then recall the basics of G.I.T..
2. A few remarks about the construction of the Hilbert scheme
Assume in this section that all schemes are Noetherian. Recall that the Hilbert
functor is a contravariant functor from schemes to sets defined as follows:
Definition 2.1. Let X � S be a projective scheme, O(1) a relatively ample line
bundle and P a fixed polynomial. Let
HilbP (X/S) : {Schemes/S} � {sets}
3
be the contravariant functor that associates to an S scheme Y the subschemes of
X ×S Y which are proper and flat over Y and have the Hilbert polynomial P .
A major theorem of Grothendieck asserts that the Hilbert functor is repre
sentable by a projective scheme.
Theorem 2.2. Let X/S be a projective scheme, O(1) a relatively ample line bundle
and P a fixed polynomial. The | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
X/S be a projective scheme, O(1) a relatively ample line bundle
and P a fixed polynomial. The functor HilbP (X/S) is represented by a morphism
HilbP (X/S) is projective over S.
u : UP (X/S) � HilbP (X/S).
I will explain some of the ingredients that go into the proof of this theorem,
leaving you to read [Gr], [Mum2], [K], [Se] and the references contained in those
accounts for complete details.
Let us first concentrate on the case X = Pn and S = Spec(k), the spectrum of a
field k. A subscheme of projective space is determined by its equations. The poly
nomials in k[x0, . . . , xn] that vanish on a subscheme form an infinite-dimensional
subvector space of k[x0, . . . , xn]. Suppose we knew that a finite-dimensional sub
space actually determined the schemes with a fixed Hilbert polynomial. Then
we would get an injection of the schemes with a fixed Hilbert polynomial into a
Grassmannian. We have already seen that the Grassmannian (together with its
tautological bundle) represents the functor classifying subspaces of a vector space.
Assuming the image in the Grassmannian is an algebraic subscheme, we can use
this subscheme to represent the Hilbert functor.
Given a proper subscheme Y of Pn and a coherent sheaf F on Y , the higher
cohomology H i(Y, F (m)), i > 0, | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
F on Y , the higher
cohomology H i(Y, F (m)), i > 0, vanishes for m sufficiently large. The finiteness
that we are looking for comes from the fact that if we restrict ourselves to ideal
sheaves of subschemes with a fixed Hilbert polynomial, one can find an integer
m depending only on the Hilbert polynomial (and not on the subscheme) that
works simultaneously for the ideal sheaf of every subscheme with a fixed Hilbert
polynomial.
Theorem 2.3. For every polynomial P , there exists an integer mP depending only
on P such that for every subsheaf I ≥ O with Hilbert polynomial P and every
integer k > mP
Pn
(1) hi(Pn, I(k)) = 0 for i > 0;
(2) I(k) is generated by global sections;
(3) H 0(Pn, I(k)) ∗ H 0(Pn , O(1)) � H 0(Pn, I(k + 1)) is surjective.
How does this theorem help? Let Y ≥ Pn be a closed subscheme with Hilbert
polynomial P . Choose k > mP . By item (2) of the theorem, IY (k) is generated by
global sections. Consider the exact sequence
0 � IY (k) � OPn (k) � OY (k) � 0.
This realizes H 0(Pn, IY (k)) as a subspace of H 0(Pn , OPn (k)). This subspace de
termines IY (k) | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
of H 0(Pn , OPn (k)). This subspace de
termines IY (k) and hence the subscheme Y . Since k depends only on the Hilbert
polynomial, we get an injection to G(P (k), H 0(Pn , OPn (k)). The image has a natu
ral scheme structure. This scheme together with the restriction of the tautological
bundle to it, represents the Hilbert functor. I will now fill in some of the details,
4
leaving most of them to you. Let us begin with a sketch of the proof of the theorem.
Definition 2.4. A coherent sheaf F on Pn is called (Castelnuovo-Mumford) m-
regular if H i(Pn , F (m − i)) = 0 for all i > 0.
Proposition 2.5. If F is an m-regular coherent sheaf on Pn , then
(1) hi(Pn , F (k)) = 0 for i > 0 and k + i → m.
(2) F (k) is generated by global sections if k → m.
(3) H 0(Pn , F (k)) ∗ H 0(Pn , O(1)) � H 0(Pn , F (k + 1)) is surjective if k → m.
Proof. The proposition is proved by induction on the dimension n. When n = 0,
the result is clear. Take a general hyperplane H and consider the following exact
sequence
0 � F (k − 1) � F (k) � FH (k) � 0.
When k = m − i, the associated long exact sequence of cohomology gives that
H i(F (m − i | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
) � 0.
When k = m − i, the associated long exact sequence of cohomology gives that
H i(F (m − i)) � H i(FH (m − i)) � H i+1(F (m − i − 1)).
In particular, if F is m-regular on Pn , then so is FH on Pn−1 . Now we can prove
the first item by induction on k. Now consider the similar long exact sequence
H i+1(F (m − i − 1) � H i+1(F (m − i)) � H i+1(FH (m − i − 1)).
The first group vanishes by induction on dimension and the third one vanishes by
the assumption that F is m regular for i → 0. We conclude that F is m + 1 regular.
Hence by induction k regular for all k > m. This proves item (1).
Consider the commutative diagram
H 0(F (k − 1)) ∗ H 0(OPn (1))
H 0(F (k − 1))
g
H 0(F (k))
u
v
H 0(FH (k − 1)) ∗ H 0(OH (1))
f
H 0(FH (k))
The map u is surjective by the regularity assumption. The map f is surjective by
induction on the dimension. It follows that v ∩ g is also surjective. Since the image
of H 0(F (k − 1)) is contained in the image of g, claim (3) follows.
It is easy to deduce (2) from (3).
�
The proof of the theorem is concluded if we can show that the ideal sheaves
of proper subchemes of Pn with a fixed Hilbert polynomial are mP -regular for an
integer depending only on P . This claim also follows by induction on the dimension
n. Choose a general hyperplane H | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
regular for an
integer depending only on P . This claim also follows by induction on the dimension
n. Choose a general hyperplane H and consider the exact sequence
0 � I(m) � I(m + 1) � IH (m + 1) � 0.
IH is a sheaf of ideals so we may use induction on the dimension.
Assume the Hilbert polynomial is given by
n
P (m) =
m
ai
i
� �
.
�
i=0
5
�
�
�
�
We then have
ψ(IH (m + 1)) = ψ(I(m + 1)) − ψ(I(m))
=
n
�
i=0
ai
��
m + 1
i
−
� � ��
m
i
n−1
=
ai+1
�
i=0
m
i
� �
Assuming the result by induction, we get an integer m1 depending only on the
coefficients a1, . . . , an such that IH has that regularity. Considering the long exact
sequence associated to our short exact sequence, we see that H i(I(m)) is isomorphic
to H i(I(m + 1) as long as i > 1 and m > m1 − i. Since by Serre’s theorem these
cohomologies vanish when m is large enough, we get the vanishing of the higher
cohomology groups. For i = 1 we only get that h1(I(m)) is strictly decreasing for
m → m1 − 1. We conclude that I is m1 + h1(I(m1 − 1))-regular. However, since I
is an ideal sheaf we can bound the latter term as follows
h1(I(m1 − 1)) = h0(I(m1 − 1)) − ψ(I(m1 − 1)) ∼ h0(OPn (m1 − 1)) − ψ(I(m1 − 1)).
This clearly depends only on the | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
1)) ∼ h0(OPn (m1 − 1)) − ψ(I(m1 − 1)).
This clearly depends only on the Hilbert polynomial; hence concludes the proof of
Theorem 2.3.
Now we indicate how one proceeds to deduce Theorem 2.2. So far we have given
an injection from the set of subshemes of Pn with a fixed Hilbert polynomial P
to the Grassmannian G(P (m), H 0(Pn , OPn (m))) for any m > mP by sending the
subscheme to the P (m)-dimensional subspace H 0(Pn, I(m)) of H 0(Pn , OPn (m))).
Of course, this subspace uniquely determines the subscheme. We still have to show
that the image has a natural scheme structure and that this subscheme represents
the Hilbert functor. For this purpose we will use flattening stratifications.
Recall that a stratification of a scheme S is a finite collection S1, . . . , Sj of locally
closed subschemes of S such that
S = S1 � · · · � Sj
is a disjoint union of these subschemes.
Proposition 2.6. Let F be a coherent sheaf on Pn × S. Let S and T be Noetherian
schemes. There exists a stratification of S such that for all morphisms f : T � S,
(1 × f )�F to Pn × T is flat over T if and only if the morphism factors through the
stratification.
This stratification is called the flattening stratification (see Lecture 8 in [Mum2]
for the details). To prove it one uses the fact that if f : X | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
Mum2]
for the details). To prove it one uses the fact that if f : X � S is a morphism of
finite type, S is integral and F is any coherent sheaf on X, then there is a dense
open subset U of S such that the restriction of F to f −1(U ) is flat over U . A
corollary is that S can be partitioned into finitely many locally closed subsets Si
such that giving each the reduced induced structure, the restriction of F to X ×S Si
is flat over Si.
We can partition S to locally closed subschemes as in the previous paragraph.
Only finitely many Hilbert polynomials Pioccur. We can conclude that there is an
integer m such that if l → m, then
and
H i(Pn(s), F (s)(l)) = 0
βS�F (l) ∗ k(s) � H 0(Pn(s), F (s)(l))
6
is an isomorphism, where βS denotes the natural projection to S.
Next one observes that (1 × f )�F is flat over T if and only if f �βS�F (l) is locally
free for all l → m. For each l we find the stratification of S such that Sl,j the
sheaf f �βS�F (l) is locally free of rank j. Note that there is the following equality
between subsets of S
≤l�mSupp[Sl,j ] = ≤m+n�l�mSupp[Sl,j ].
This is because the Hilbert polynomials have degree at most n.
For each integer h → 0, | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
Sl,j ].
This is because the Hilbert polynomials have degree at most n.
For each integer h → 0, there is a well-defined locally closed subscheme of S
defined by
≤0�r�hSr,Pi (m+r).
When h → n, these form a decreasing sequence of subschemes with the same sup
port. Therefore, they stabilize. These give us the required stratification.
The flattening stratification allows us to put a scheme structure on the image of
our map to the Grassmannian. More precisely, consider the incidence correspon
dence
I ≥ Pn × G(P (mP ), H 0(Pn , OPn (mP ))).
The incidence correspondence has two projections
β1 : I � Pn
and
β2 : I � G(P (mP ), H 0(Pn , OPn (mP ))).
For the rest of this section we will abbreviate G(P (mP ), H 0(Pn , OPn (mP ))) simply
by G. β�
2 T (−mP ) where T is the tautological bundle on G is an idea sheaf of OPn ×G.
Let us denote the corresponding subscheme by Y . The flattening stratification of
OY over G gives a subscheme HP of G corresponding to the Hilbert polynomial P .
(Note that this is the scheme structure that we put on the set we earlier obtained.)
The claim is that HP represents the Hilbert functor and the universal family is the
restriction W of Y to the inverse image of HP .
Suppose we have a subscheme X ≥ Pn × S mapping to S via f and flat over S
(and suppose the Hilbert polynomial is P ). We obtain an exact sequence
0 � f�IX (mP ) � f�OPn×S (mP ) � f�OX (mP | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
obtain an exact sequence
0 � f�IX (mP ) � f�OPn×S (mP ) � f�OX (mP ) � 0.
By the universal property of the Grassmannian G, this induces a map g : S � G.
Since
f�IX (m) = g �β2�IY (m)
for m sufficiently large, we see that (1 × g)�OY is flat with Hilbert polynomial
P , hence g factors through HP by the definition of the flattening stratification.
Moreover, X is simply S ×HP W . This concludes the construction of HilbP (Pn/S).
Exercise 2.7. Verify the details of the above construction.
So far we have constructed the Hilbert scheme as a quasi-projective subscheme
of the Grassmannian. To prove that it is projective it suffices to check that it is
proper. This is done by checking the valuative criterion of properness. This follows
from the following proposition [Ha] III.9.8.
7
Proposition 2.8. Let X be a regular, integral scheme of dimension one. Let p ⊗ X
be a closed point. Let Z ≥ Pn
X−p be a closed subscheme flat over X − p. Then there
exists a unique closed subscheme Z ⊗ Pn flat over X, whose restriction to Pn
is
X
Z.
X−p
Exercise 2.9. Deduce from the proposition that the Hilbert scheme we constructed
is projective.
Exercise 2.10. For a projective scheme X/S construct HilbP (X/S) as a locally
closed subscheme of HilbP (Pn/S).
Exercise 2.11. Suppose X and Y | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
locally
closed subscheme of HilbP (Pn/S).
Exercise 2.11. Suppose X and Y are projective schemes over S. Assume X is
flat over S. Let Hom(X, Y ) be the functor that associates to any S scheme T the
set of morphisms
X ×S T � Y ×S T.
Using our construction of the Hilbert scheme and noting that a morphism may be
identified with its graph construct a scheme that represents the functor Hom(X, Y ).
2.1. Examples of Hilbert schemes. In this subsection we would like to give
some explicit examples of Hilbert schemes.
Example 2.12. Consider the Hilbert scheme associated to a projective variety X
and the Hilbert polynomial 1. Then the Hilbert scheme is simply X.
Exercise 2.13. Show that if C is a smooth curve, then Hilbn(C) is simply the
symmetric n-th power of C. In particular, Hilbn(P1) = Pn
Exercise 2.14. Show that the Hilbert scheme of hypersurfaces of degree d in Pn
is isomorphic to P(n+d)−1 .
Example 2.15 (The Hilbert scheme of conics in P3). Any degree 2 curve is nec
essarily the complete intersection of a linear and quadratic polynomial. Moreover,
the linear polynomial is uniquely determined. We thus obtain a map
d
Hilb2n−1(P3) � P3� .
The fibers of this map are Hilb2n−1(P2) which is isomorphic to P5 . We conclude by
Zariski’s main theorem that that Hilb2n−1(P3) is the P5 bundle P(Sym2T �) � P3� .
Of course, in all this discussion we needed the fact that Hilb2 | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
P5 bundle P(Sym2T �) � P3� .
Of course, in all this discussion we needed the fact that Hilb2n−1(P3) is reduced.
Theorem 2.16. Let X be a projective scheme over a field k and Y ≥ X be a closed
subscheme, then the Zariski tangent space to Hilb(X) at [Y ] is naturally isomorphic
to HomY (IY /I 2
Y , OY ).
In particular, in our case the dimension of T Hilb2n−1(P3) = h0(NC/P3 ) = 8.
Hence Hilb2n−1(P3) is reduced (in fact smooth). Hilb2n−1(P3) is one of the few
examples where we can answer many of the geometric questions we can ask about
a Hilbert scheme.
We can use the Hilbert scheme of conics to solve the following question:
Question 2.17. How many conics in P3 intersect 8 general lines in P3?
As in the case of Schubert calculus, we can try to calculate this number as an
intersection in the cohomology ring. The cohomology ring of a projective bundle
over a smooth variety is easy to describe in terms of the chern classes of the bundle
and the cohomology ring of the variety.
8
Theorem 2.18. Let E be a rank n vector bundle over a smooth, projective variety
ci(E)ti . Let α denote the
X. Suppose that the chern polynomial of E is given by
first chern class of the dual of the tautological bundle over PE. The cohomology of
PE is isomorphic to
�
H �(PE) �
=
H �(X) [α]
< α n | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
is isomorphic to
�
H �(PE) �
=
H �(X) [α]
< α n + α n−1c1(E) + · · · + cn(E) = 0 >
If you are not familiar with chern classes, see the handout about chern classes.
Using Theorem 2.18 we can compute the cohomology ring of Hilb2n−1(P3). Recall
that T � on P3� is a rank 3 vector bundle with chern polynomial
c(T �) = 1 + h + h2 + h3 .
Using the splitting principle we assume that the polynomial splits into three linear
factors
(1 + x)(1 + y)(1 + z).
Then the chern polynomial of Sym2(T �) is given by
(1 + 2x)(1 + 2y)(1 + 2z)(1 + x + y)(1 + x + z)(1 + y + z).
Multiplying this out and expressing it interms of the elementary symmetric poly
nomials in x, y, z, we see that
c(Sym 2(T �)) = 1 + 4h + 10h2 + 20h3 .
It follows that the cohomology ring of Hilb2n−1(P3) is given as follows:
H �(Hilb2n−1(P3)) �
=
Z[h, α]
< h4 , α 3 + 4hα 2 + 10h2α + 20h3 >
The class of the locus of conics interseting a line is given by 2h + α. This can be
checked by a calculation away from codimension at least 2. Consider the locus of
planes in P3� that do not contain the line l. Over this locus there is a line bundle
that associates to each point (H, Q) on Hilb | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
l. Over this locus there is a line bundle
that associates to each point (H, Q) on Hilb2n−1(P3) the homogeneous quadratic
polynomials modulo those that vanish at H ≤ l. This line bundle is none other
than the pull-back of OP3� . The tautological bundle over Hilb2n−1(P3) maps by
evaluation. The locus where the evaluation vanishes is the locus of conics that
intersect l. Hence the class is the difference of the first chern classes. Finally, we
compute (2h + α)8 using the presentation of the ring to obtain 92.
Over the complex numbers we can invoke Kleiman’s theorem to deduce that
there are 92 smooth conics intersecting 8 general lines in P3 .
Exercise 2.19. Calculate the number of conics that intersect 8 − 2i lines and
contain i points for 0 ∼ i ∼ 3.
Exercise 2.20. Calculate the class of conics that are tangent to a plane in P3 .
Find how many conics are tangent to a general plane and intersect 7 general lines.
Exercise 2.21. Generalize the previous discussion to conics in P4 . Calculate the
numbers of conics that intersect general 11 − 2i − 3j planes, i lines and j points.
Example 2.22 (The Hilbert scheme of twisted cubics in P3). The Hilbert poly
nomial of a twisted cubic is 3t + 1. This Hilbert scheme has two components.
A general | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
twisted cubic is 3t + 1. This Hilbert scheme has two components.
A general point of the first component parameterizes a smooth rational curve of
degree 3 in P3 . A general point of the second component parameterizes a degree
9
3 plane curve together with a point in P3 . Note that the dimension of the first
component is 12, whereas the dimension of the second component is 15. Hence
the Hilbert scheme is not pure dimensional. The component of the Hilbert scheme
parameterizing the smooth rational curves has been studies in detail. In fact, that
component is smooth.
Exercise 2.23. Describe the subschemes of P3 that are parameterized by the com
ponent of the Hilbert scheme that parameterizes smooth rational curves of degree
3 in P3 .
Piene and Schlessinger proved that the component of the Hilbert scheme pa
rameterizing twisted cubics is smooth. In analogy with our analysis of the Hilbert
scheme of conics we can try to compute invariants of cubics using the Hilbert
scheme. Unfortunately, this turns out to be very difficult.
Problem 2.24. Calculate the number of twisted cubics intersecting 12 general
lines in P3 .
Problem 2.25. Calculate the number of twisted cubics that are tangent to 12
general quadric hypersurfaces in P3 . (Hint: There are 5,819,539,783,680 of them.)
Towards the end of the course we will see how to use the Kontsevich moduli space
to answer these questions.
Unfortunately, Hilbert schemes are | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
the end of the course we will see how to use the Kontsevich moduli space
to answer these questions.
Unfortunately, Hilbert schemes are often unwieldy schemes to work with. They
often have many irreducible components. It is hard to compute the dimensions of
these components. Even components of the Hilbert scheme whose generic point
parameterizes smooth curves in P3 may be everywhere non-reduced.
Example 2.26 (Mumford’s example). Mumford showed that there exists a com
ponent of the Hilbert scheme parameterizing smooth curves of degree 14 and genus
24 in P3 that is non-reduced at the generic point of that component. See [Mum1]
or [HM] Chapter 1 Section D.
The pathological behavior of most Hilbert schemes make them hard to use for
studying the explicit geometry of algebraic varieties. In fact, the Hilbert schemes
often exhibit behavior that is arbitrarily bad. For instance, R. Vakil recently proved
that all possible singularities occur in some component of the Hilbert scheme of
curves in projective space.
Theorem 2.27 (Murphy’s Law). Every singularity class of finite type over SpecZ
occurs in a Hilbert scheme of curves in some projective space.
3. Basics about curves
Here we collect some basic facts about stable curves.
If β : C � S is a stable curve of genus g over a scheme S, then C has a relative
dualizing sheaf �C/S with the following properties
(1) The formation of �C/S commutes with base change.
(2) If S = Spec k where k is an algebraically closed field and C is the normal
ization of C, then �C/S may be identified with the | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
is the normal
ization of C, then �C/S may be identified with the sheaf of meromorphic
differentials on C that are allowed to have simple poles only at the inverse
image of the nodes subject to the condition that if the points x and y lie
over the same node then the residues at these two points must sum to zero.
˜
˜
10
(3) In particular, if C is a stable curve over a field k, then H 1(C, � C/k ) = 0
if n → 2 and � ∗n
is very ample for n → 3. When n = 3 we obtain a
C/k
tri-canonical embedding of stable curves to P5g−6 with Hilbert polynomial
P (m) = (6m − 1)(g − 1).
∗n
To see the third property observe that every irreducible component E of a stable
curve C either has arithmetic genus 2 or more, or has arithmetic genus one but
meets the other components in at least one point, or has arithmetic genus 0 and
meets the other components in at least three points. Since �C/k ∗ OE is isomorphic
to �E/k (
i Qi) where Qi are the points where E meets the rest of the curve. Since
this sheaf has positive degree it is ample on each component E of C, hence it is
i Qi) has positive degree on each component, hence �1−n ∗ OE has
ample. �E/k(
no sections for any n → 2. By Serre duality, it follows that H 1(C, � C/k ) = | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
n → 2. By Serre duality, it follows that H 1(C, � C/k ) = 0. To
show that when n → 3, �C/k is very ample, it suffices to check that �C/k separates
points and tangents.
C/k
∗n
�
�
∗n
∗n
Exercise 3.1. Check that when n → 3, �C/k separates points and tangents.
∗n
4. Stable reduction
Stable reduction was originally proved by Deligne and Mumford using the ex
istence of stable reduction for abelian varieties [DM]. [HM] Chapter 3 Section C
contains a beautiful account which we will summarize below.
The main theorem is the following:
Theorem 4.1 (Stable reduction). Let B be the spectrum of a DVR with function
field K. Let X � B be a family of curves with n sections χ1, . . . , χn such that
the restriction XK � Spec K is an n-pointed stable curve. Then there exists a
finite field extension L/K and a unique stable family X � B ×K L with sections
χn such that the restriction to Spec L is isomorphic to XK ×K L.
˜
χ1, . . . , ˜
˜
One can algorithmically carry out stable reduction (at least in characteristic
zero). Since stable reduction is an essential tool in algebraic geometry we begin by
giving some examples. We will then sketch the proof.
Example 4.2. Fix a smooth curve C of genus g → 2. Let p ⊗ C be a fixed point
and | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
Fix a smooth curve C of genus g → 2. Let p ⊗ C be a fixed point
and let q be a varying point. More precisely, we have the family C × C � C with
two sections χp : C � C × C mapping a point q to (q, p) and χq : C � C × C
mapping q to (q, q). All the fibers are stable except when p = q. To obtain a stable
family, we blow up C × C at (p, p). The resulting picture looks as follows (see
Figure 2):
There is an algorithm that produces the stable reduction in characteristic zero.
This algorithm is worth knowing because the explicit calculation of the stable limit
often has applications to geometric problems.
Step 1. Resolve the singularities of the total space of the family. The result of
this step is a smooth surface X mapping to our initial surface. Moreover, we can
assume that the support of the central fiber is a normal-crossings divisor.
Step 2. After Step 1 at every point of the central fiber the pull-back of the
uniformizer may be expressed as xa for some a > 0 at a smooth point or xayb for
11
q
p
p
q
Figure 2. Stable reduction when two marked points collide.
a pair a, b > 0 at a node. Make a base change of order p for some prime dividing
the multiplicity of a multiple component of the fiber.
Step 3. Normalize the resulting surface.
Suppose the central fiber was of the form
i niCi The effect of doing steps 2
and 3 is to take | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
of the form
i niCi The effect of doing steps 2
and 3 is to take a branched cover of the surface X branched along the reduction of
the divisor forming the central fiber modulo p. Repeat steps 2 and 3 until all the
components occuring in the central fiber appear with multiplicity 1.
�
Step 4. Contract the rational components of the central fiber that are not stable.
Sketch of proof of Theorem 4.1. We will assume that n = 0 and then make some
remarks about how to modify the statements here to obtain the general case. Let
R be a DVR with uniformizer z. Let φ ⊗ B = Spec R be the generic point. We
are assuming that our family X� is a stable curve of genus g.
Consider regular, proper B-schemes that extend X� . By results of Abhyankar
[Ab] about resolutions of surface singularities there exists a unique relatively mini
mal model of X� . Consider the completion of the local ring at a node of the special
fiber. This ring is isomorphic to R[[x, t]]/(xy −zn) for some integer n → 1. This ring
is not regular for n > 1. We can desingularize it in a sequence of ∈n/2◦ blow-ups.
Over the node we get a sequence of −2-curves.
Let X be a proper, flat regular surface extending X� . Let Ci, i = 1, . . . , n, be
the components of the special fiber. Suppose they occur with multiplicity ri. Recall
the following basic facts about the components of the special fiber
(1) The special fiber C is connected and the multiplicities ri > 0 for all i. | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
special fiber
(1) The special fiber C is connected and the multiplicities ri > 0 for all i.
(2) Ci · Cj → 0 for all i �= j and Ci · C = 0 for all i.
(3) If K is the canonical class, then the arithmetic genus of Ci is given by the
genus formula as
C 2 + Ci · K
.
2
(4) The intersection matrix Ci ·Cj is a negative definite symmetric matrix. The
aiCi with the property that Z 2 = 0 are
1 +
i
only linear combinations Z =
rational multiples of C.
�
One can divide the components Ci of the special fiber into the following categories
�
12
Example 4.3. Suppose we have a general pencil of smooth curves of genus g in
P2 specializing to a curve with an ordinary m-fold point. We may write down the
equation of such a family as F + tG where G is the equation defining a general
curve of genus g and F locally has the form
m
(y − aix) + h. o. t.
�
i=1
with distinct ai. To perform stable reduction we blow-up the m-fold point. In the
resulting surface the proper transform C of the central fiber is smooth of genus
g − m(m − 1)/2, but the exceptional divisor is a P1 that meets C in m points and
occurs with multiplicity m. We make a base change of order m. We get an m-fold
cover of this P1 totally ramified at the m points of intersection with C. By the
Riemann-Hurwitz formula this is a genus m(m − | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
points of intersection with C. By the
Riemann-Hurwitz formula this is a genus m(m − 3)/2 + 1. The stable limit then is
as shown in the figure.
Exercise 4.4. Suppose Ct is a general pencil of smooth genus g plane curves
acquiring an ordinary cusp (a singularity whose local equation is given by y2 = x3).
Describe the stable limit of this family of curves.
Exercise 4.5. Read and do the exercises in Chapter 3 Section C of [HM].
5. Deligne-Mumford Stacks
In this section for completeness I will give you the definition of Deligne-Mumford
stacks. I will summarize a few basic results and definitions. Much better accounts
exist in [DM], [Ed] and [LM-B]. See also [Fan].
Let S be the category of schemes over a scheme S. A category T over S is a
category together with a functor p : T � S.
Definition 5.1 (Groupoid). A category ((T, p) over S is a groupoid if the following
two conditions hold
(1) If f : B≤ � B is a morphism in S and C is an object in T lying over B,
then there exists an object C ≤ over B≤ and a morphism ζ : C ≤ � C such
that p(ζ) = f .
(2) Let C, C ≤, C ≤≤ be objects in T lying over the objects B, B≤, B≤≤ in S, re
spectively. If ζ : C ≤ � C and ω : C ≤≤ � C are morphisms in T | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
: C ≤ � C and ω : C ≤≤ � C are morphisms in T and
f : B≤ � B≤≤ is a morphism in S satisfying p(ω) ∩ f = p(ζ), then there is a
unique morphism π : C ≤ � C ≤≤ such that ω ∩ π = ζ and p(π ) = f .
Example 5.2. Recall that a Deligne-Mumford stable curve (or simply a stable
curve) of genus g → 2 over a scheme S is a proper, flat family β : C � S whose
geometric fibers are reduced, connected, one dimensional schemes Cs satisfying the
following properties:
(1) The only singularities of Cs are ordinary double points.
(2) A non-singular rational component of Cs meets the other components in at
least three points.
(3) Cs has arithmetic genus g—equivalently h1(OCs ) = g.
We can define a groupoid Mg of Deligne-Mumford stable curves of genus g over
schemes over Spec Z as follows: The sections of Mg over a scheme X are families
13
of stable curves C � X. A morphism between C ≤ � X ≤ and C � X is a fiber
diagram
C ≤
C
X
which induces an isomorphism C ≤ �= X ≤ ×X C.
� �
X ≤
Mg is a groupoid and it is the main example that we are interested in.
For the sake of future constructions and definitions it is important to keep in
mind the examples of two more groupoids.
Example 5.3. Any contravariant functor F : S � {sets} from schemes | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
groupoids.
Example 5.3. Any contravariant functor F : S � {sets} from schemes to sets
gives rise to a groupoid (usually also called F by abuse of notation). The objects
of the groupoid F are pairs (X, �) where X is a scheme and � is an element of the
set F (X). A morphism between (X, �) and (Y, λ) is a morphism f : X � Y such
that F (f )(λ) = �. In particular, this construction allows us to view schemes as
groupoids. To a scheme X we can associate its functor of points Hom(�, X). Since
this is a contravariant functor from schemes to sets, to a scheme X we can also
associate a groupoid X. The distinction between a scheme X and the associated
groupoid is often blurred.
Example 5.4. Since the construction of many moduli spaces involves taking the
quotient of a parameter space (such as a component of a Hilbert scheme) by a
group action, the groupoid [X/G] is important. Let X be a scheme and G a group
scheme acting on X. The sections of [X/G] over a scheme Y are principal G-bundles
E � Y together with a G-equivariant map E � X. A morphism between two such
principal G-bundles is a pull-back diagram.
Exercise 5.5. There is a relation between the previous two examples. Show that
if the action of G on X is free and a quotient scheme X/G exists, then then there
is an equivalence of categories between [X/G] and the groupoid associated to the
scheme X/G.
Let (T, p) be a group | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
] and the groupoid associated to the
scheme X/G.
Let (T, p) be a groupoid. For any two objects X and Y in the fiber of T over
a scheme B, we can associate a functor IsomB (X, Y ). This functor associates to
any morphism f : B≤ � B, the set of isomorphisms in T (B≤) between f �(X) and
f �(Y ).
In the case of Deligne-Mumford stable curves, given any two stable curves C and
C ≤ , IsomX (C, C ≤) associates to any morphism f : Y � X the set of isomorphisms
between f �(C) and f �(C ≤). Recall that C and C ≤ are both canonically polarized
by �C/X and �C /X , respectively. Moreover, the formation of the relative dual
izing sheaf commutes with base change. Consequently, any isomorphism satisfies
f �(�C /X ) = �C/X . Hence, all isomorphisms are isomorphisms between polarized
schemes. It follows by the existence of the Hilbert scheme, that IsomX (C, C ≤) is
represented by a scheme quasi-projective over X.
Definition 5.6 (Stack). A groupoid (T, p) over S is a stack if
(1) IsomB (X, Y ) is a sheaf in the ´etale topology for all B, X and Y ;
14
�
�
(2) If {Bi � B} is a covering of B in the ´etale topology, and Xi are a collection
of objects in T (Bi | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
{Bi � B} is a covering of B in the ´etale topology, and Xi are a collection
of objects in T (Bi) with isomorphisms
ζi,j : Xj|Bi ×B Bj � Xi|Bi ×B Bj
in T (Bi ×B Bj ) satisfying the cocycle condition, then there exists an object
X ⊗ T (B) with isomorphisms X|Bi � Xi inducing the isomorphisms ζi,j .
Example 5.7. The groupoid [X/G] defined in Example 5.4 is a stack. Let e, e≤ be
two objects in [X/G](Y ) corresponding to two principal G-bundles E, E ≤ � Y with
G-equivariant maps f, f ≤ to X, respectively. IsomY (e, e≤) is empty unless E = E≤
and f = f ≤ . In the latter case the isomorphisms correspond to the subgroup of
G that stabilizes the map f . Since the functor that associates to a G-equivariant
map its stabilizer is representable, condition (1) follows. Condition (2) also holds
for principal G-bundles.
Let Pg,n(m) be the Hilbert polynomial (2nm − 1)(g − 1), the Hilbert polynomial
of an n-canonically embedded stable curve. Set N = n(2g−2)−g. Let H g,n the sub-
scheme of the Hilbert scheme Hilb(2nm−1)(g−1)(PN ) parameterizing n-canonically
embedded stable curves. Below we will show that there is an equivalence of cate
gories between Mg and [H g,n/PGL(N + 1)] where the action of PGL(N + 1) on
the Hilbert scheme is the one induced | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
where the action of PGL(N + 1) on
the Hilbert scheme is the one induced by its usual action on PN . In particular, it
follows from the previous example that Mg is a stack.
Recall in example 5.3 we associated to a scheme a groupoid. Observe that this
groupoid is a stack. The second condition is satisfied because the functor of points
of a scheme is represented by the scheme itself. In particular, we can view each
scheme as a stack. In the litterature stacks that arise this way are usually referred
to as schemes meaning that the stack associated to the scheme. We will also indulge
in this habit.
A morphism of stacks f : T � T ≤ is representable if for any map of a scheme
X � T ≤ the fiber product T ×T X is represented by a scheme. We can transport
the notions of morphisms of schemes to representable morphisms of stacks in the
following way: We say that a representable morphism f : T � T ≤ has a property P
(such as quasi-compact, separated, proper, etc.) if for all maps of a scheme X � T ≤ ,
the corresponding morphism of schemes T ×T X � X has the property P .
Definition 5.8 (Deligne-Mumford stack). A stack is called a Deligne-Mumford
stack if
(1) The diagonal ΩX : T � T ×S T is representable, quasi-compact and sepa
rated;
(2) There exists a scheme U and an ´etale, surjective morphism U � T .
Morphisms as in condition (2) are called ´etale atlases.
The following is a useful theorem for verifying that | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
� T .
Morphisms as in condition (2) are called ´etale atlases.
The following is a useful theorem for verifying that a stack is a Deligne-Mumford
stack (see [DM] Theorem 4.21, or [Ed] Theorem 2.1).
Theorem 5.9. Let T be a quasi-separated stack over a Noetherian scheme S.
Suppose that
(1) The diagonal is representable and unramified,
(2) There exists a scheme U of finite type over S and a smooth, surjective
S-morphism U � F .
15
The F is a Deligne-Mumford stack.
A consequence of this theorem is that if X/S is a Noetherian scheme of finite
type and G/S is a smooth group scheme acting on X with with finite and reduced
stabilizers, then [X/G] is a Deligne-Mumford stack. The conditions on the stabiliz
ers (that they are finite and reduced) guarantee that IsomB (E, E) are unramified.
It follows that the diagonal is unramified. The second condition in the theorem is
satisfied by the map X � [X/G].
Given the equivalence of categories between Mg and [H g,n/PGL(N + 1)] it
follows that Mg is a Deligne-Mumford stack because the action of PGL(N + 1) on
H g,n has finite and reduced stabilizers.
Just like in the case of schemes there are valuative criteria for separatedness and
properness. We now state these and observe that Mg is a proper Deligne-Mumford
stack. For the following two theorems let f : T � S be a morphism of finite type
from a | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
For the following two theorems let f : T � S be a morphism of finite type
from a Deligne-Mumford stack to a noetherian scheme S
Theorem 5.10 (The valuative criterion for separatedness). The morphism f is
separated if and only if for any complete discrete valuation ring with algebraically
closed residue field and any commutative diagram
T
�� � � � � � � � �
g1 ,g2
f
S
Spec R
any isomorphism between the restrictions of g1 and g2 to the generic point of Spec R
can be extended to an isomorphism of g1 and g2.
Theorem 5.11 (The valuative criterion of properness). If f is separated, then f
is proper if and only if, for any discrete valuation ring R with field of fractions
K and any map Spec R � T which lifts over Spec K to a map to T , there is a
finite extension K ≤ of K such that the lift extends to all of Spec R≤ where R≤ is the
integral closure of R in K ≤ .
The stable reduction theorem together with the valuative criterion of properness
implies that Mg is a proper Deligne-Mumford stack.
One approach for constructing the coarse moduli scheme (which we cannot com
plete at present because we have not yet developed the theory of divisors on the
moduli stack) is to first construct the moduli space as an algebraic space, then ex
hibit an ample divisor on the coarse moduli algebraic space. This approach has been
applied successfully to represent many moduli functors. The first step | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
moduli algebraic space. This approach has been
applied successfully to represent many moduli functors. The first step is achieved
by a corollary of a general theorem of Keel and Mori [KM] (see also [Li] for a nice
treatment).
Theorem 5.12. Any separated Deligne-Mumford stack of finite type has a coarse
moduli space in the category of algebraic spaces.
Once we study the ample cone in the Picard group of the moduli stack, we
will be able to deduce the existence of a coarse moduli scheme from the previous
theorem. The second approach to the construction of the coarse moduli scheme is to
directly take the G.I.T. quotient of the Hilbert scheme parameterizing n-canonically
embedded stable curves. The advantage of the first approach is that it does away
16
�
�
�
with delicate calculations describing the stable and semi-stable loci of this action.
The first approach may also be used to construct moduli spaces in other situations.
The advantage of the second approach is that it produces a projective coarse moduli
scheme at once.
6. The GIT construction of the moduli space
Good references for this section are [HM] Chapter 4, [Mum3], [FKM] and [Ne].
Explaining the GIT construction in detail would take us too far afield. Instead we
will briefly sketch the main ideas and refer you to the literature.
6.1. Basics about G.I.T.. An algebraic group G is a group together with the
structure of an algebraic variety such that the multiplication and inverse maps are
morphisms of varieties. An action of an algebraic group G on a variety X is | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
isms of varieties. An action of an algebraic group G on a variety X is a
morphism f : G × X � X such that f (gg≤, x) = f (g, f (g≤, x)) and f (e, x) = x,
where e is the identity of the group. The stabilizer of a point x ⊗ X is the closed
subgroup of G fixing x. The orbit of a point x under G is the image of f restricted
to G × {x}.
For our purposes we can always restrict attention to SL(n), GL(n) or PGL(n).
An algebraic group which is isomorphic to a closed subgroup of GL(n) is called
a linear algebraic group. A group is called geometrically reductive if for every
linear action of G on kn and every non-zero invariant point v ⊗ kn, there exists an
invariant homogeneous polynomial that does not vanish on v. The group is called
linearly reductive if the homogeneous polynomial may be taken to have degree
one. Finally a group is called reductive if the maximal connected normal solvable
subgroup is isomorphic to a direct product of copies of k� . In characteristic zero
these concepts coincide. In characteristic p > 0 a threorem of Haboush guarantees
that every reductive group is geometrically reductive.
The question is to obtain a quotient of a variety under the action of a reductive
group.
Lemma 6.1. Let G be a geometrically reductive group acting on an affine variety
X. Let W1 and W2 be two disjoint invariant closed orbits. Then there exists an
invariant polynomial f ⊗ A(X) | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
two disjoint invariant closed orbits. Then there exists an
invariant polynomial f ⊗ A(X)G such that f (W1) = 0 and f (W2) = 1.
Proof. Pick any h ⊗ A(X) such that h(W1) = 0 and h(W2) = 1. Consider the
subspace spanned by hg for g ⊗ G. This is a finite dimensional subspace. To see
this consider the function H(g, x) = h(gx) in A(G × X) � A(G) ∗ A(X). We
=
can write H(g, x) as a finite sum
Fi ∗ Hi in A(G) ∗ A(X) of the generators of
A(G) and A(X). Hence the subspace spanned by hg for g ⊗ G is contained in the
subspace spanned by the Hi. Pick a basis for this subspace h1, . . . , hn. We obtain
a rational representation of G on this subspace, hence a linear action on kn making
the morphism β : X � kn given by β(x) = (h1(x), . . . , hn(x)) into a G-morphism.
Since G is geometrically reductive there is an invariant polynomial f that has the
value zero on β(W1 ) and the value 1 on β(W2). f ∩ β is the desired polynomial. �
�
i
The main theorem for quotients of reductive group actions on affine varieties is
the following:
17
Theorem 6.2. Let G be a reductive group acting on an affine variety X. Then
there exists a quotient affine variety Y and a G-invariant, surjective morphism
ζ : X � Y such that
(1 | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
variety Y and a G-invariant, surjective morphism
ζ : X � Y such that
(1) For any open set U ≥ Y , the ring homomorphism
ζ� : A(U ) � A(ζ−1 (U ))
is an isomorphism of A(U ) with A(ζ−1(U ))G .
(2) If W ≥ X is a closed invariant subset, then ζ(W ) is closed in Y .
(3) If W1 and W2 are disjoint closed invariant sets, then their images under ζ
are disjoint.
Proof. The main technical results are provided by a theorem of Haboush and a
theorem of Nagata.
Theorem 6.3 (Haboush). Any reductive group G is geometrically reductive.
Theorem 6.4 (Nagata). Let G be a geometrically reductive group acting rationally
on a finitely generated k-algebra R. Then the ring of invariants RG is finitely
generated.
In view of these theorems A(X)G is finitely generated. Hence we can let Y =
Spec A(X)G . The inclusion of A(X)G � A(X) induces a morphism ζ : X � Y .
�
The claimed properties are easy to check for ζ.
Remark 6.5. The following are straightforward observations:
(1) For any open subset U ≥ Y , (U, ζ) is a categorical quotient of ζ−1(U ) by
G.
(2) The images of two points in X coincide if and only if the orbit closures of
these two points intersect. Consequently, | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
) The images of two points in X coincide if and only if the orbit closures of
these two points intersect. Consequently, Y will be an orbit space if and
only if the orbits of the G action on X are closed.
Remark 6.6. We will not prove Haboush’s theorem here. The interested reader
may consult the original paper [Hab]. Over the complex numbers reductive, ge
ometrically reductive and linearly reductive coincide. This follows from the fact
that any finite dimensional representation is decomposible to irreducible represen
tations. Projection to the one-dimensional invariant subspace produces the desired
invariant linear functional.
We now sketch the proof of Nagata’s theorem. Since R is a finitely generated
k-algebra, we can pick generators f1, dots, fn that generate R. We can also assume
that the subspace spanned by the fi is G-invariant. (If not, we can replace it by
a minimal G-invariant subspace, which is finite-dimensional by the argument in
Lemma 6.1.) We thus obtain a linear G action on the subspace spanned by fi by
setting
Let S = k[X1, . . . , Xn]. There is an action of G on S by setting
f g =
i
�
j
�i,j (g)fj .
X g =
i
�
j
�i,j (g)Xj .
There is a k-algebra homomorphism from S to R sending Xi to fi that is compatible
with the G actions. We are thus reduced to proving Nagata’s theorem in the case
18
when G acts on S preserving degree, Q ≥ S is a G-invariant ideal with the | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
ata’s theorem in the case
18
when G acts on S preserving degree, Q ≥ S is a G-invariant ideal with the induced
action on R = S/Q. Under these assumptions we would like to see RG is finitely
generated.
Suppose not. Since S is Noetherian, there exists an ideal Q maximal among those
that are G-invariant such that RG where R = S/Q is not finitely generated. Then
if J �= 0 is a G-invariant homogeneous ideal in R, then (R/J)G is finitely generated.
Suppose first there is a homogeneous ideal Q with the desired properties.
I claim that (R/J)G is integral over RG/(J ≤ RG). Suppose f ⊗ (R/J)G . Pick
h ⊗ R such that the image of h in R/J is f . We would like to find h0 ⊗ RG such
that (h)t − h0 for some positive integer t is in RG . Look at the finite-dimensional,
G-invariant subsapce M generated by hg . [Unfortunately, there is potential for
confusion between hg and (h)t . The first denotes the g-translate of h, the second
denotes the t-th power of h. To distinguish between these two, we will put paren
theses around h in the latter case.] Since J is invariant, hg − h is in J for every g.
We conclude that M ≤ J has codimension 1 in M . We can write every element in
M uniquely as ah + h≤ where a ⊗ k and h≤ ⊗ M ≤ J. Sending ah + h≤ to a defines
a G-invariant linear functional l on M .
There is an action of G also on M � . If we let h, | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
on M .
There is an action of G also on M � . If we let h, j2, . . . , jn be a basis of M
where ji ⊗ M ≤ J, we can identify M � with kr in terms of the dual basis. The
linear functional l corresponds to the vector (1, 0, . . . , 0). Since G is geometrically
reductive, there exists an invariant homogeneous polynomial F ⊗ k[X1, . . . , Xn] of
t does not vanish. Consider the morphism
degree t → 1 such that the coefficient of X1
k[X1, . . . , Xn] sending X1 to h and Xi to ji for i > 1.If h0 is the image of F , ht − h0
belongs to J. We conclude that (R/J)G is integral over RG/(J ≤ RG).
If A is a finitely generated k-algebra which is integral over a subalgebra B, then
B is finitely generated. Hence in our case, RG/(J ≤ RG) is finitely generated. In
fact, (R/J)G is a finite RG/(J ≤ RG)-module.
Choose a non-zero homogeneous element f of RG of degree at least one. If
f is not a zero-divisor, f R ≤ RG = f RG . Since RG/f RG is finitely generated,
(RG/f RG)+ is finitely generated as an ideal. Hence RG is finitely generated as an
+
ideal in RG . Hence RG is a finitely generated k-algebra.
Exercise 6.7. Modify the last paragraph of the proof in case f is a zero-divisor.
Hint: Consider the homogeneous | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
the last paragraph of the proof in case f is a zero-divisor.
Hint: Consider the homogeneous ideal I of elements of R that annihilate f . Since
RG/(f R ≤ RG) and RG/I ≤ RG are both finitely generated, there is a finitely
generated subalgebra of RG that surjects onto both these algebras
In order to handle the non-homogeneous case, we may assume that RG is a
domain. By the homogeneous case S G is finitely generated. RG is integral over
SG/Q≤SG . It suffices to show that the field of fractions of RG is a finitely generated
extension of k. Let T be the set of non-zero divisors of R. Form the ring of fractions
of R with respect to T . Let m be the maximal ideal. The field of fractions of RG
may be identified with a subfield of T −1R/m. Since T −1R/m is the field of fractions
of the finitely generated k-algebra R/m ≤ R, this follows.
Example 6.8. Everyone’s favorite example is the action of GL(n) on the space of
n × n matrices Mn by conjugation. The space of matrices is isomorphic to affine
space An . Hence, the coordinate ring is k[ai,j ], 1 ∼ i, j ∼ n. Any conjugacy class
19
2
has a representative in Jordan canonical form which is unique upto a permutation
of the Jordan blocks. Since the set of eigenvalues of a matrix is invariant under
conjugation, we see that the elementary symmetric polynomials of the eigenval
ues, i.e. the coefficients | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
the elementary symmetric polynomials of the eigenval
ues, i.e. the coefficients of the characteristic polynomial, are invariant under the
action. Conversely, suppose that a polynomial is invariant under conjugation. If
the eigenvalues are distinct, we can diagonalize the matrix by connjugation. Hence
the polynomial must be a symmetric function of the eigenvalues. If the eigenvalues
are repeated, the diagonal matrix is in the closure of the orbits with non-trivial
Jordan blocks. We conclude that any invariant polynomial is a symmetric polyno
mial of the eigenvalues. Since the elementary symmetric polynomials generate the
ring of symmetric polynomials, we conclude that the ring of invariant functions is
generated by the coefficients of the characteristic polynomial.
Now we would like to extend the discussion from actions of reductive groups on
affine varieties to actions on projective varieties. Suppose we have a group acting
on a projective variety X ≥ Pn . A linearization of the action of G is a linear action
on kn+1 which induces the given action on X. More generally, let X be a variety,
G a group acting on it and L a line bundle on X. A linearization of the action of
G with respect to L is a linear action on L that induces the action of G on X.
Definition 6.9. A point x ⊗ X is called semi-stable if there exists an invariant
homogeneous polynomial that does not vanish on x. A point x ⊗ X is called stable
if there exists an invariant polynomial f that does not vanish on x, the action of
G on Xf is closed and the dimension of the orbit of x is equal to the dimension of
G. These depend not only on the action, but the chosen linearization. Denote the | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
the orbit of x is equal to the dimension of
G. These depend not only on the action, but the chosen linearization. Denote the
locus of semi-stable points by X ss and the locus of stable points by X s .
Remark 6.10. Note that the semi-stable points are precisely those that do not
contain 0 in the closure of their orbits. Both X ss and X s are clearly open (possibly
empty) in X.
The main theorem of G.I.T. is the existence of a good quotient of the semi-stable
locus whose restriction to the stable locus is a geometric quotient. We will call a
quotient a good quotient if it satisfies the conditions of Theorem 6.2. We will call
a good quotient that is also an orbit space a geometric quotient.
Theorem 6.11. Let X be a projective variety in Pn . Then for every linear action
of a reductive group G on X
(1) There exists a good quotient (Y, ζ) of X ss by G and Y is projective.
(2) There exists an open subset Y s of Y such that ζ−1(Y s) = X s and (Y s, ζ)
is a geometric quotient of X s .
In view of this theorem it is important to determine the stable and semi-stable
loci for reductive group actions on projective varieties. Unfortunately, this in gen
eral is a very difficult problem. There is one instance where stability and semi-
stability is easy to determine.
Definition 6.12. A one-parameter subgroup is a homomorphism � : Gm � G.
Any action of k� on kn+1 can be | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
a homomorphism � : Gm � G.
Any action of k� on kn+1 can be diagonalized. Hence, there exists a basis
e0, . . . , en such that the action of the one-parameter subgroup � is given by �(t)ei =
20
tri ei for some integers ri. If ˆ =
x
xiei, then
Define
�
x�(t)ˆ =
tri xiei.
�
i
µ(x, �) = max{−ri | xi �= 0 }.
Theorem 6.13 (The Hilbert-Mumford criterion of stability). Let G be a reductive
group acting linearly on a projective variety X ≥ Pn . Then:
(1) x is semi-stable if and only if for every one-parameter subgroup � of G
µ(x, �) → 0.
(2) x is stable if and only if for every one-parameter subgroup � of G µ(x, �) >
0.
Proof. The challenging part of the theorem is to produce a one-parameter subgroup
that has the wrong µ invariant if x is not semi-stable. We will sketch Hilbert’s proof
for the case G = SL(m). The general case follows the same general line of argument
(see §2.1 [FKM]).
g /
x where ˆ
Let K be the field of fractions of R = k[[T ]]. If x is not stable, then the morphism
G � kn+1 given by sending g to gˆ
x is a lift of x is not proper. By the
g ⊗ SL(m, K) such that ¯x ⊗ Rn+1 ,
valuative criterion of properness, there exists ¯
but ¯ ⊗ SL(m, R). We can, however, clear denominators so that T rg¯ ⊗ SL | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
�
but ¯ ⊗ SL(m, R). We can, however, clear denominators so that T rg¯ ⊗ SL(m, R)
for some r. The ring R is a P.I.D., hence we can decompose ¯ = g¯1dg¯2 where g1 and
g2 are in SL(m, R) and d is a diagonal matrix consisting of entries T w1 , . . . , T wm
for some integers wi whose sum is zero (since the resulting matrix has to be in
SL(m, K). This is the point in the proof where we are using that G = SL(m). To
prove the theorem for general groups one needs to use a theorem of Iwahori which
asserts that the double coset in G(R)\G(K)/G(R) for a reductive group can be
represented by a one-parameter subgroup.
gˆ
g
Let g2 be the matrix obtained by setting T = 0 in g¯2. The de-stabilizing one-
parameter subgroup is defined by
�(t) = g −1diag(tw1 , . . . , twm )g2.
2
Diagonalize the action of � on kn+1 with respect to a basis e0, . . . , en as above.
We would like to show that if ˆ = 0, then the weight ri of the action on ei is
non-negative. We can also consider the basis e0, . . . , en as a basis of K n+1 . Then
g −1dg2ei = T ri ei. In particular,
2
−1 g¯ = g
−1 g¯1dg¯2 = (g −1dg2)g
xi �
g¯1
−1
g
2
Therefore, the i-th component | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
g −1dg2)g
xi �
g¯1
−1
g
2
Therefore, the i-th component of g −1 g¯1
−1
g
2
in R, we conclude that ri → 0.
−1
2
g¯1
2
¯ x. Consequently, the i-th component of g −1 g2 ˆ
g2 ˆ
2
gˆ
g¯2.
−1 ¯x is T ri times the i-th component of
¯ x is in T −ri R. Since it is also
�
2
−1
2
Exercise 6.14. Modify the previous argument to obtain the theorem for the semi-
stable case.
Example 6.15 (Points on P1). Consider the action of SL(2) on the homogeneous
polynomials of degree d in two variables. Let � be a one-parameter subgroup of
SL(2). If we diagonalize the action of � on k2 by diag(ta, t−a) in coordinates
(x, y), then the monomials xiyd−i diagonalizes the action of � on homogeneous
21
polynomials of degree d. The weight of the action on xiyd−i is a(2i − d). If we
want the weight to be negative, then the coefficient of one monomial xiy
with
2i − d < 0 has to be non-zero. This means that a homogeneous polynomial is
stable if and only if it does not have any zeros with multiplicity → d/2. Similarly,
a homogeneous polynomial is semi-stable if and only if it does not have any zeros
with multiplicity > d/2.
d−i | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
is semi-stable if and only if it does not have any zeros
with multiplicity > d/2.
d−i
i
j
Example 6.16 (Cubic plane curves). Consider the action of SL(3) on the homo
geneous polynomials of degree 3 in three variables. If we diagonalize the action
of a one-parameter subgroup � in terms of the coordinates x1, x2, x3 such that
�(t)xi = twi xi, then the basis given by monomials x1x2x 3−i−j diagonalizes the
action of � on degree 3 homogeneous polynomials. The weight of the action on
j 3−i−j
i x2x
is given by iw1 + jw2 + (3 − i − j)w3. We can visualize the one param
x1
3
eter subgroup in terms of barycentric coordinates. The one-parameter subgroups
correspond in this picture to lines pivoted around the point (i, j, 3−i−j) = (1, 1, 1).
If we move the line without crossing any integral points on the triangle, we do not
change the conditions for stability. Also the picture is invariant under the sym
metries of the triangle. Analyzing the coefficients we see that a cubic is stable if
and only if it is smooth. Similarly a cubic is semi-stable if and only if it has ordi
nary double points. Note that the G.I.T. quotient of the stable locus in this case
constructs the j-line.
3
Exercise 6.17. Try to generalize the previous example to the action of SL(3 | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
constructs the j-line.
3
Exercise 6.17. Try to generalize the previous example to the action of SL(3) on
homogeneous polynomials of degree 4, 5, 6, .... In particular, describe what kinds of
singularities are allowed on stable curves of degree 4, 5, 6...
6.2. The construction of M g . In view of Theorem 6.11 in order to construct M g
we need to show that the N -canonically embedded Deligne-Mumford stable curves
are stable points for the SL(n + 1)-action on the Hilbert scheme and that they form
a closed subset. The details of this verification are involved. You may find good
accounts in [HM] and [Mum3].
We would like to apply the Hilbert-Mumford criterion to the action of SL(n + 1)
on HilbP (m)(Pn). Fix a one-parameter subgroup � of SL(n + 1). Suppose in
terms of homogeneous coordinates xi that diagonalize the action, the weights are
w0, . . . , wn. Of course, as usual we have that
i wi = 0. Recall that we exhibited
the Hilbert scheme as a subscheme of the Grassmannian G(P (m), H 0(Pn , OPn (m)))
for m greater than or equal to the regularity of all the ideal sheaves with Hilbert
polynomial P . The Grassmannian has natural Pl¨ucker coordinates consisting of
P (m)-element subsets of monomials in the xi of degree m. This basis also diag
P (m) H 0(Pn , OPn (m)). The weight on the
onalizes the action of SL(n + 1) on
Pl¨
ucker coordinate { Yj1 , . . | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
the
onalizes the action of SL(n + 1) on
Pl¨
ucker coordinate { Yj1 , . . . , YjP (m) } where Yji =
is given by
mji ,r
xr
r
�
�
wr mji ,r.
�
i,r
The Hilbert-Mumford criterion for semi-stability then translates to the condition
that for each one parameter subgroup, there should be a non-vanishing Pl¨ucker
coordinate whose weight is non-positive.
We begin by showing that the m-th Hilbert points of smooth, non-degenerate
curves embedded by a complete linear series of degree d → 2g are stable for the
SL(n + 1) action.
22
Theorem 6.18 (Stability for smooth curves). Let C be a smooth curve of genus
g → 2 embedded in projective space Pd−g by a complete linear system of degree d
at least 2g. Then C is Hilbert stable. Moreover, there exists M such that for all
m → M , the m-th Hilbert point of non-degenerate, smooth curves of degree d and
genus g in Pd−g is stable.
Sketch. The proof is an application of the Hilbert-Mumford criterion.
�
Definition 6.19 (Potential stability). A connected curve C of degree d and genus
g in Pd−g+1 is called potentially stable if
(1) The embedded curve C is non-degenerate.
(2) The abstract curve C is moduli semi-stable.
(3) The linear series embedding C is complete and non-special ( | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
(2) The abstract curve C is moduli semi-stable.
(3) The linear series embedding C is complete and non-special (i.e. has h1 = 0).
(4) If C ≤ is a complete subcurve of C of arithmetic genus g≤ meeting the rest
of the curve C in k points, then the following estimate holds
degC (OC (1)) −
d
g − 1
k
(gC − 1 + )
2
k
.
2
∼
�
�
�
�
�
�
�
�
Remark 6.20. Observe that if C ≤ is a smooth rational curve meeting the rest of
the curve in exactly two points (k = 2), then the term gC − 1 + k/2 = 0, hence the
degree of C ≤ has to be 1. In other words, C ≤ is a line. By the same argument, if C ≤ is
a nodal tree of smooth rational curves meeting the rest of C in exactly two points,
then C ≤ is a smooth rational curve since the degree is at most one. Furthermore,
C ≤ cannot meet the rest of the curve in only one point.
Recall that �C|C is the dualizing sheaf �C twisted by the nodes connecting C ≤
to C. Hence, deg(�C|C ) = 2gC − 2 + k. Condition (4) has the following alternative
useful expression
deg C ≤ − d
�
�
�
�
deg(�C|C )
deg(�C )
�
�
�
�
∼
k
.
2
Theorem 6.21 (Potential stability). Let g → 2 and d > 9(g − 1). Then there is
an integer M depending only on | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
2 and d > 9(g − 1). Then there is
an integer M depending only on d and g such that if m → M and C ⊗ Pd−g is a
connected curve with semi-stable m-th Hilbert point, then C is potentially stable.
The proof of this theorem is quite lengthy eventhough the strategy is straight
forward. We suppose C has a geometric property that violates potential stability.
Under this assumption we construct a one-parameter subgroup that destabilizes
the Hilbert point of C contradicting the assumption that the m-th Hilbert point of
C was semi-stable.
We first assume Theorem 6.21 and deduce from it the existence of the coarse
moduli space M g . Fix an integer r → 5. Consider r-canonically embedded stable
∗r
is very ample for r → 3, every Deligne-Mumford stable curve
curves. Since �C
has a representative in the Hilbert scheme H = Hilbr(2g−2)+1−g(Pr(2g−2)−g ). Now
consider the subscheme H of H subscheme of the Hilbert scheme parameterizing
r-canonically embedded Deligne-Mumford stable curves. Let H ss denote the inter-
section of H with the semi-stable locus of H. Since r → 5, we have that the degree
of the curves are at least 10(g − 1) > 9(g − 1). Therefore, the assumption of the
Potential Stability Theorem is satisfied. We conclude that every semi-stable point
ˆof H is potentially stable.
ˆ
ˆ
ˆ
23 | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
satisfied. We conclude that every semi-stable point
ˆof H is potentially stable.
ˆ
ˆ
ˆ
23
Lemma 6.22. The locus H ss is closed in semi-stable locus of the Hilbert scheme
H ssˆ .
Proof. To show that H ss is closed we need to show that the inclusion H ss � ˆ
H ss
is proper. By the valuative criterion of properness it suffices to check that given
a map from the spectrum of a DVR to ˆH ss whose generic point lies in H ss, the
closed point also lies in H ss . Given such a map consider the universal curve CR over
Spec (R). There are two line bundles on CR, the relative dualizing sheaf �CR/R and
OCR (1). These two are isomorphic except possibly at the central fiber. To conclude
the lemma we need to show that they also agree on the central fiber. Hence the
i aiCi is a linear combination of the central
two differ by OCR (−
fiber. We need that ai = 0 for all i. We can assume that ai → 0 for all i with at
≤ be the subcurve of the central fiber D where ai > 0 and
least one ai = 0. Let C1
C ≤ be the subcurve of the central fiber D where ai = 0. We see that all ai = 0 as
follows. A local equation of OCR (−
i aiCi) is identically zero on every component
≤ . In particular, the local equation vanishes at the
≤ and on no component of C1
of C2
k | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
every component
≤ . In particular, the local equation vanishes at the
≤ and on no component of C1
of C2
k points of intersection between C1
≤ . We then have that
�
≤ and C2
i aiCi) where
�
�
2
k ∼ degD (OCR (−
�
i
aiCi) ∼ degD (OCR (1)) −
degC (OCR (1)|C )
2
degC
2
(�|C
2
)
2 degD (�|C
2
) ∼
k
.
2
�
Lemma 6.23. Every curve C whose Hilbert point lies in H ss is Deligne-Mumford
stable.
Proof. By the potential stability theorem C is semi-stable. In order to show that
it is stable we need to check that there are no rational curves that intersect the
rest of the curve in only two points. On a rational curve meeting the rest of C in
two points, the degree of the dualizing sheaf of C is zero whereas OC (1) is very
ample. Since these two coincide for points in H ss, we conclude that C must be
�
Deligne-Mumford stable.
Lemma 6.24. Every Deligne-Mumford stable curve of genus g has a model in H ss .
Proof. Every moduli stable curve C is embedded in Pr(2g−2)−g by its �
∗r
. We need
C
to show that the Hilbert point of C lies in H ss . If C is smooth, we already know this
by Theorem 6.18. To deduce it for singular Deligne-Mumford stable curves, we take
a one-parameter deformation of C to a smooth curve of genus g over the spectrum
of | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
singular Deligne-Mumford stable curves, we take
a one-parameter deformation of C to a smooth curve of genus g over the spectrum
of a DVR R. If we embed this curve r-canonically, we get a map from Spec R
to the Hilbert scheme. The generic point lies in H ss . Since the G.I.T. quotient
of the Hilbert scheme ˆH ss by the action of the special linear group is projective,
after a base change we can extend the map to ˆH ss . Since H ss is closed, the image
of the map lies in H ss . Pulling back the universal curve we obtain a semi-stable
reduction of a family of stable curves. By the uniqueness of semi-stable reduction
this family has to agree with our original family. Since the curves H ss are actually
stable, the central fiber of both families have to be projectively equivalent. The
�
lemma follows.
Lemma 6.25. Every curve whose Hilbert point lies in H ss is Hilbert stable.
Proof. We need to show that every point in H ss has closed orbit and the stabilizer
of a point in H ss is finite. Suppose the stabilizer is not finite, then the curve
24
C would have infinitely many automorphisms contradicting that Deligne-Mumford
stable curves have only finitely many automorphisms. If the orbit is not closed, then
the closure would contain a semi-stable orbit with positive dimensional stabilizer.
�
Again we would obtain a contradiction.
Lemma 6.26. The locus H ss is non-singular.
C proper and flat over Spec k | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
Lemma 6.26. The locus H ss is non-singular.
C proper and flat over Spec k[[t1, . . . , tr ]] where r = dim Ext1(�1
Proof. Recall that given a Deligne-Mumford stable curve C, there exists a formal
scheme ˜
C , OC ) such
that the special fiber is isomorphic to C. Moreover, for a stable curve the versal
deformation is universal and algebrizable and the generic fiber is smooth.
˜
Let [C] ⊗ H ss be a point. Let C be the universal formal deformation of C over
B = Spec k[[t1, . . . , tr]]. Set S be the formal completion of H ss at [C]. By the
universal property of the Hilbert scheme we get a map S � H ss . By the universal
property there exists a unique morphism f : S � B such that the pull-back of the
universal curve is S ×B C. The Lemma follows from the claim that f : S � B is
�
formally smooth.
˜
One important aspect of the G.I.T. construction is that the projectivity of M g
is immediate. Another important consequence is the irreducibility of the moduli
space of curves over an algebraically closed field of any characteristic. Originally
Deligne and Mumford developed the theory of Deligne-Mumford stacks to prove
the irreducibility in all characteristics and for all genus in [DM].
Theorem 6.27. The moduli space M g is projective.
Theorem 6.28. The moduli space M g is irreducible (and reduced) over any | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
6.28. The moduli space M g is irreducible (and reduced) over any alge
braically closed field.
Proof. Soon we will see that the moduli space of curves in characteristic zero is
irreducible. There are many ways of seeing this. We will use Teichm¨uller theory
to construct Mg as the quotient of a bounded, contractible domain in C3g−3 . Al
ternatively, one can exhibit every smooth curves as a branched cover of P1 . When
the number of branch points is large relative to the degree of the map, using the
combinatorics of the symmetric group one may show that the space of branched
covers of P1 is irreducible. Suppose now that the characteristic of the field k is
positive. Let R be a discrete valuation ring whose quotient field has characteris
tic zero and whose residue field is k. The construction outlined so far works over
R /PGL � Spec R is connnected, by Zariski’s
Spec R. Since the generic fiber of H ss
R /PGL ∗ k is connected. Since this is an orbit space H ss
connectedness theorem H ss
is connected. Since it is smooth, it is reduced and irreducible. Consequently M g is
also irreducible. M g is also reduced because the structure sheaf of the quotient is
�
the sheaf of invariants of the structure sheaf of H ss .
k
Finally we enumerate the steps that one carries out in order to prove the Poten
tial Stability Theorem. We assume that a | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
enumerate the steps that one carries out in order to prove the Poten
tial Stability Theorem. We assume that a geometric condition violating potential
stability occurs on a curve. We then produce a one-parameter subgroup destabiliz
ing that point, hence showing that it is not a Hilbert stable point. Unfortunately
the number of cases and calculations needed to give a complete proof is rather large.
Since we will not use these techniques later in the course, we will just sketch a few
sample cases. A complete proof can be found on pages 35-87 of [G].
25
Claim 6.29. The first claim is that if a curve C is Hilbert stable, then Cred is not
contained in a hyperplane.
If the curve is degenerate, then the map H 0(OPn (1)) � H 0(Cred,
OCred (1)) has non-trivial kernel. Use the filtration that assigns weight −1 to sec
tions vanishing on Cred and weight w > 0 to the others so that the average weight is
0. There exists an integer q such that the q-th power of the ideal sheaf of nilpotents
in OC is zero. Hence no monomial that contains more than q factors of weight
−1 can be zero. Provided we choose m such that (m − q)w > q, every element of
a monomial basis of H 0(C, OC (m)) has positive weight. Hence, C is not Hilbert
semi-stable. From now on we may assume that the linear span of our curves in
Pn . This argument is the blueprint for the other arguments. We will give very few
details for the other ones.
Claim 6.30. The second claim is that every component of C is generically reduced.
Claim | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
details for the other ones.
Claim 6.30. The second claim is that every component of C is generically reduced.
Claim 6.31. The third claim is that every singularity of Cred is a double point.
If p is a point of multiplicity 3 or more, the two-step filtration assigning weight
0 to the sections vanishing at p and weight one to the others is destabilizing.
Claim 6.32. Every double point of Cred is a node.
Claim 6.33. H 1(Cred, OC (1)) = 0
Claim 6.34. C is reduced.
From these claims it follows that the first three conditions of the definition of
potential stability hold. The final step is to show that the estimate in (4) holds.
This is done by showing that if not the filtration FC is destabilizing.
References
[Ab]
[DM]
[Ed]
[Fan]
[G]
[Gr]
S. S. Abhyankar. Resolution of singularities of arithmetical surfaces. In Arithmetical
Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), pages 111–152. Harper & Row,
New York, 1965.
P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus.
IHES Publ. Math. 36(1969), 75–110.
D. Edidin. Notes on the construction of the moduli space of curves. In Recent progress
in intersection theory (B | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
the construction of the moduli space of curves. In Recent progress
in intersection theory (Bologna, 1997), Trends Math., pages 85–113. Birkh¨auser Boston,
Boston, MA, 2000.
B. Fantechi. Stacks for everybody. In European Congress of Mathematics, Vol. I
(Barcelona, 2000), volume 201 of Progr. Math., pages 349–359. Birkh¨auser, Basel, 2001.
D. Gieseker. Lectures on moduli of curves, volume 69 of Tata Institute of Fundamen
tal Research Lectures on Mathematics and Physics. Published for the Tata Institute of
Fundamental Research, Bombay, 1982.
A. Grothendieck. Techniques de construction et th´ emes d’existence en g´
emas de Hilbert. In S´
alg´
221, 249–276. Soc. Math. France, Paris, 1995.
etrie
eminaire Bourbaki, Vol. 6, pages Exp. No.
ebrique. IV. Les sch´
eom´
eor`
[Hab] W. J. Haboush. Reductive groups are geometrically reductive. Ann. of Math. (2)
[HM]
[Ha]
[KM]
[K]
102(1975), 67–83.
J. Harris and I. Morrison. Moduli of curves. Springer-Verlag, 1998.
R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
Springer-Verlag, 1998.
R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in
Mathematics, No. 52.
S. Keel and S. Mori. Quotients by groupoids. Ann. of Math. (2) 145(1997), 193–213.
J. Koll´ar. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik
und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-
Verlag, Berlin, 1996.
26
[LM-B] G. Laumon and L. Moret-Bailly. Champs alg´ebriques, volume 39 of Ergebnisse der Math
ematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics
[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in
Mathematics]. Springer-Verlag, Berlin, 2000.
M. Lieblich. Groupoids and quotients in algebraic geometry. In Snowbird lectures in
algebraic geometry, volume 388 of Contemp. Math., pages 119–136. Amer. Math. Soc.,
Providence, RI, 2005.
[Li]
[Mum1] D. Mumford. Further pathologies in algebraic geometry. Amer. J. Math. 84(1962), 642–
648.
[Mum2] D. Mumford. Lectures on curves on an algebraic surface. With a section by G. M.
Bergman | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
on curves on an algebraic surface. With a section by G. M.
Bergman. Annals of Mathematics Studies, No. 59. Princeton University Press, Princeton,
N.J., 1966.
[Mum3] D. Mumford. Stability of projective varieties. Enseignement Math. (2) 23(1977), 39–110.
[FKM] D. Mumford, J. Fogarty, and F. Kirwan. Geometric invariant theory, volume 34. Springer-
[Ne]
[Se]
Verlag, Berlin, third edition, 1994.
P. E. Newstead. Introduction to moduli problems and orbit spaces, volume 51 of Tata
Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute
of Fundamental Research, Bombay, 1978.
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Pure and Applied Mathematics. Queen’s University, Kingston, ON, 1986.
27 | https://ocw.mit.edu/courses/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006/06c4e979c3dadcba7d0f3dd69efdb315_const.pdf |
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6.641 Electromagnetic Fields, Forces, and Motion, Spring 2005
Please use the following citation format:
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6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7: Polarization and Conduction
I. Experimental Observation
A. Fixed Voltage - Switch Closed (v = V o )
As an insulating material enters a free-space capacitor at constant voltage
more charge flows onto the electrodes; i.e. as x increases, i increases.
B. Fixed Charge - Switch open (i=0)
As an insulating material enters a free space capacitor at constant charge,
the voltage decreases; i.e. as x increases, v decreases.
II. Dipole Model of Polarization
A. Polarization Vector P = N p = N q d
( p = q d dipole moment)
N dipoles/Volume ( P is dipole density)
d
+ q
−q
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 1 of 27
Courtesy of Krieger Publishing. Used with permission.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 2 of 27
Cite as: Markus Zahn, course materials for 6.641 Electromagnetic Fields, Forces, and Motion, Spring 2005.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
agnetic Fields, Forces, and Motion, Spring 2005.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
Q
inside V = − (cid:118)∫
q N d
S
i
da
= ∫ ρP dV
V
paired charge or
equivalently
polarization
charge density
Qinside V = − P i da = − ∇ i P dV =
(cid:118)∫
S
∫
V
ρ dV
P
∫
V
(Divergence Theorem)
P = q N d
∇ i P = −ρP
B. Gauss’ Law
∇ i (ε E) = ρ
o
total
= ρ + ρ = ρ − ∇ i P
u
P
u
unpaired charge
density; also
called free charge
density
∇ i (ε
o
+ ) = ρ
E P
u
D = εo E + P
Displacement Flux Density
∇ i D = ρu
C. Boundary Conditions
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 3 of 27
Da − Db ⎤
∇ i D = ρu ⇒ (cid:118)∫ D i da = ∫ ρu dV ⇒ n i ⎡
⎣
⎦ = σ su
S
V
P a − Pb ⎤
∇ i P = −ρ P ⇒ (cid:118)∫ P i da = −∫ ρP dV ⇒ n i ⎡
⎣
⎦ = −σ sp
S
V
∇ i (ε E) = ρ + ρ ⇒ (cid:118)∫ ε E i da = ∫ (ρ + ρ ) dV ⇒ n i ε ⎣
u
u
o
o
o
P
P | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
ρ + ρ ) dV ⇒ n i ε ⎣
u
u
o
o
o
P
P
⎡Ea − Eb ⎦
⎤ = σ + σ
su
sp
S
V
D. Polarization Current Density
∆Q = q N dV
=
q N d da
i
=
P
i da
[Amount of Charge passing through
surface area element da ]
d ip =
∂∆Q
∂t
=
∂P
∂t
i da
= Jp i da
polarization current density
[Current passing through surface
area element da ]
Jp =
∂P
∂t
Ampere’s law:
∇ x H = Ju + Jp + εo
∂E
∂t
= Ju +
∂P
∂t
+ εo
∂E
∂t
= Ju +
∂ (ε E + P)
∂t
o
= Ju +
∂ D
∂t
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 4 of 27
III. Equipotential Sphere in a Uniform Electric Field
)
lim Φ r, θ = − E r cos θ
r → ∞
(
o
⎡Φ = − E z = −E r cos θ⎤
⎣
⎦
o
o
Φ (r = R,
θ) = 0
Φ (r, θ) = − Eo
⎡
⎢r −
⎣
3 ⎤
R
2
r ⎦
⎥ cos θ
This solution is composed of the superposition of a uniform electric field
plus the field due to a point electric dipole at the center of the sphere:
Φ
dipole
=
p cos θ
4πε ro
2
with
p =
4
πε E
o o
3
R
This dipole is due to the surface charge distribution on the sphere.
σ (r | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
4
πε E
o o
3
R
This dipole is due to the surface charge distribution on the sphere.
σ (r = R,
s
θ) = ε E (r = R,
o r
θ) = − ε ∂Φ
o ∂r r R
=
=
ε oEo
⎡
⎢1 +
⎣
3
2R
3
r
⎤
⎥ cos
r R ⎦=
θ
= εo o
3 E cos θ
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 5 of 27
IV. Artificial Dielectric
E =
v
d
, σ = ε E =
s
ε v
d
q = σsA =
ε A
d
v
C =
q
v
=
ε A
d
E
d
ε
_
υ
+
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
For spherical array of non-interacting spheres (s >> R)
P = 4 π ε R 3 E
i z ⇒ P = N p = 4 π ε R 3 E N
o
o
o
z
o
z
_
N = 1
3
s
⎡
⎛
P = εo ⎢4 π ⎜
⎝
⎢
⎣
R
s
3 ⎤
⎞
⎟ ⎥ E = ψ e εo E
⎠ ⎥
⎦
⎛
⎛
⎜ ψ e = 4 π ⎜
⎜
⎝
⎝
3 ⎞
R
⎞
⎟ ⎟
⎟
s
⎠
⎠
ψe (electric susceptibility)
D = εo E + P = ε o ⎡⎣1 + ψ e ⎤⎦ E = ε E
εr (relative dielectric constant)
ε
ε=
r
ε
o
ε =
⎡1 + ψ ⎤ = | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
εr (relative dielectric constant)
ε
ε=
r
ε
o
ε =
⎡1 + ψ ⎤ = ε
e ⎦
o ⎣
⎛
⎛
⎜1 + π 4
⎜
o ⎜
⎝
⎝
R
s
3 ⎞
⎞
⎟
⎟ ⎟
⎠ ⎠
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 6 of 27
V. Demonstration: Artificial Dielectric
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 7 of 27
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
E = ⇒ σ = ε E =
s
v
d
ε v
d
q = σsA =
ε A
d
v ⇒ C = =
q
v
ε A
d
∆i = ω ∆C V =
v
o
R s
∆C =
( − o ) A
ε
ε
d
⎛
= 4 π εo ⎜
⎝
⎞R
⎟
⎠s
3
A
d
R=1.87 cm, s=8 cm, A= (0.4)2 m2, d=0.15m
ω =2π(250 Hz), Rs=100 k Ω , V=566 volts peak
∆ C=1.5 pf
v = ω ∆C R
V
s
0
=(2π) (250) (1.5 x 10-12) (105) 566 = 0.135 volts
peak
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 8 of 27
VI | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
.6 x 10-19 Coulombs, m-=9.1 x 10-31 kg
n-=1020/m3 , ε = ε o ≈ 8.854 x 10 −
12
farads/m
ω =
p−
2q n
−
m − ε
−
≈ 5.6 x 10
11
rad/s
fp − =
ωp − ≈ 9 x 10
2π
10 Hz
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 9 of 27
B. Drift-Diffusion Conduction [Neglect inertia]
0
m +
+dv
dt
= q
E
+
− m ν v −
+
+
+
+
∇ (n k T )
n+
⇒ v =
+
q
m +ν +
+ E −
k T
m + ν +n+
∇n
+
0
m −
dv
−
dt
= −q E
−
J = q +
+ n v+ =
+
− m ν v −
−
−
−
)
( −
∇ n k T
n−
⇒ v =
−
−q− E −
m −ν −
k T
m −ν −n−
∇n
−
2
+
q n+
m +ν +
E − +
q k T
m +ν +
∇n+
J =
−
−q− n− v− =
−
2q n
m −ν −
− E + −
q k T
m −ν −
∇n−
ρ + = q+ n+ , ρ − = −q− n−
J = ρ µ E − D ∇ρ
+
+
+
+
+
J =
−
−ρ − µ − E − D −∇ρ −
µ | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
�ρ
+
+
+
+
+
J =
−
−ρ − µ − E − D −∇ρ −
µ + =
q+
m +ν +
µ − =
q−
m −ν −
,
,
D + =
k T
m + ν +
D − =
k T
m − ν −
charge mobilities
Molecular
Diffusion
Coefficients
D + =
µ
+
D − =
µ
−
k T = thermal voltage (25 mV @ T ≈ 300o K)
q
Einstein’s Relation
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 10 of 27
C. Drift-Diffusion Conduction Equilibrium (J+ = J− = 0)
J = 0 = ρ µ E − D ∇ρ = −ρ µ ∇Φ − D ∇ρ
+
+
+
+
+
+
+
+
+
J = 0 = −ρ µ E − D ∇ρ =
−
ρ µ ∇Φ − D ∇ρ
−
−
−
−
−
−
−
−
∇Φ = −
D + ∇ρ + =
ρ µ
+
+
−k T
q
∇ (ln
ρ + )
∇Φ =
D
−
ρ µ
−
−
∇ρ − =
k T
q
∇ (ln
ρ − )
ρ + = ρoe q / kT
− Φ
Boltzmann Distributions
ρ = −ρ e q / kT
o
+ Φ
−
ρ Φ(
+
= 0
) = ρ Φ = 0 ) = ρ
(
−
−
o
[Potential is zero when system is charge neutral]
2
∇ Φ =
−ρ
ε
= −
+
(ρ + ρ− )
ε
=
−ρo
ε ⎣ | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
−x / L d
e
x > 0
−
ε V
L d
o e+x / L d
2
x < 0
σs (x = 0 ) = ε ⎡⎣Ex (x = 0+ ) − Ex (x = 0− )⎤⎦ =
ε
2 Vo
Ld
2. Point Charge (Debye Shielding)
2
∇ Φ −
Φ
2 = 0
L d
1 ∂ ⎛ 2 ∂Φ ⎞
r
r2 ∂r ⎜
∂r ⎟
⎠
⎝
=
1 ∂2
r ∂r
2 (r Φ)
E. Ohmic Conduction
⇒
d2
)
(
dr2 r Φ −
r Φ
L2
d
= 0
0
r Φ = A e
1
−r / Ld
+ A e
2
+r / L d
Φ ( )r =
Q
4 π ε r
−r / L d
e
J = ρ µ E
+
+
+
− D +
∇ρ
+
J = −ρ µ E
−
−
−
− D −
∇ρ
−
If charge density gradients small, then ∇ρ ± negligible ⇒ ρ + = −ρ − = ρ o
J = J+ + J− = (ρ µ − ρ µ ) E = ρ (µ + µ ) E = σE
−
−
o
+
+
−
+
J = σ E (Ohm’s Law)
σ = ohmic conductivity
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 13 of 27
F. pn Junction Diode
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 14 of 27
∆Φ = Φ − Φ | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
Electrodes:
R =
l
Aσ
,
ε
A
C =
l
⇒
ε
RC =
σ
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 16 of 27
Coaxial
R =
ln b
a
2 π σ l
, C =
2 π ε l
b
ln
a
ε
⇒ RC =
σ
Concentric Spherical
1
R 1
−
1
R 2
4 π σ
R =
, C =
4 π ε
1
1
−
R 1 R 2
ε
⇒ RC =
σ
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 17 of 27
VIII. Change Relaxation in Uniform Conductors
∇ i Ju +
∂ ρ
∂ t
u = 0
ρ
∇ i E =
u
ε
J =
u
σ E
σ ∇ i E
+
∂ ρ
∂ t
u = 0
⇒
∂ ρ
u
∂ t
+
σ
ε
ρ = 0
u
ρ
u
ε
∂ ρu
∂ t
+
τ =
e
σε = dielectric relaxation time
ρu = 0 ⇒ ρ = ρ (r, t = 0 ) e − t τ
0
u
τ
e
e
IX. Demonstration 7.7.1 – Relaxation of Charge on Particle in Ohmic Conductor
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
Courtesy of Hermann A. Haus and James R. Melcher. Used with permission.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 18 of 27
(cid:118)
∫ J da = | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
Zahn
Lecture 7
Page 18 of 27
(cid:118)
∫ J da =
i
σ (cid:118)
∫ E da =
i
S
S
σ qu
ε
=
−dq
dt
dq q
dt
τe
+ = 0 ⇒ q = q t = 0 e − τ
t
e
)
(
(τe =
σ)
ε
Partially Uniformly Charged Sphere
Courtesy of Krieger Publishing. Used with permission.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 19 of 27
ρu (t = 0) =
ρ0
r R 1<
0
r > R 1
Q T =
4
3
π R 1
3 ρ0
( )
ρu t =
ρ e t
0
− τ
e
r < R
1
(τ = ε σ)
e
0
r > R 1
− τ
ρ0 r e t e
3 ε
=
Q r e t e
− τ
4 π ε R 3
1
0 <
r
< R 1
r (
E r, t
) =
Q e t e
− τ
4 π ε r 2
Q
4 π ε0 r 2
R <
1
r R
<
2
r > R
2
σ su
)
(
r = R = ε0 E r = R 2 − ε E r = R 2 −
r (
2 )
+ )
r (
=
Q
4 R 2π
2 (1 e t e
− − τ )
X. Self-Excited Water Dynamos
A. DC High Voltage Generation (Self-Excited)
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 20 of 27 | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 20 of 27
From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
Courtesy of Herbert Woodson and James Melcher. Used with permission. Woodson, Herbert H., and James R. Melcher.
Electromechanical Dynamics, Part 2: Fields, Forces, and Motion. Malabar, FL: Kreiger Publishing Company, 1968. ISBN: 9780894644597.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 21 of 27
− n C v = C
1
i
−n C v = C
2
i
dv2
dt
dv1
dt
v = V (cid:108) est
1
1
⇒
⇒
− n C V (cid:108) = C s V (cid:108)
i
1
2
v = V (cid:108) est
2
2
− n C V (cid:108) = C s V (cid:108)
1
i
2
⎡n Ci
⎢ Cs
⎢
⎢
⎢
⎢
⎢
⎣
1
⎤ ⎡ (cid:108) ⎤
V1 ⎥
1 ⎥ ⎢
⎥ ⎢
⎥
⎥ = 0
⎥ ⎢
⎥ ⎢
⎥
n Ci
⎥
⎢⎣V(cid:108)
⎥⎦
⎥
Cs
⎦
2
Det = 0
n C i
C
⊕ root blows up
n Ci t
e C
Any perturbation grows exponentially with time
2
⎛ n Ci ⎞
⎜
⎝ Cs ⎠
⎟ = 1 ⇒ s = ±
B. AC High Voltage Self | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
Ci ⎞
⎜
⎝ Cs ⎠
⎟ = 1 ⇒ s = ±
B. AC High Voltage Self – Excited Generation
−n C v = C
1
i
−n C v = C
2
i
−n C v = C
3
i
dv2
dt
dv3
dt
dv1
dt
;
v = V e
1
(cid:108)
1
st
v = V e
2
(cid:108)
2
st
v = V (cid:108) est
3
3
1
V(cid:108)
Cs
⎤
0 ⎤ ⎡
⎥
⎢
⎥ ⎢
⎥
⎥ ⎢
⎥
⎥ ⎢V(cid:108)
⎥
⎥ ⎢
⎥
⎥ ⎢
⎥
⎥
V(cid:108)
n C i
3 ⎥
⎦ ⎢
⎦
⎣
2
= 0
⎡n Ci
⎢
⎢
⎢0
⎢
⎢
⎢
⎣Cs
Cs
n C
i
0
det = 0
From Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 22 of 27
(n Ci )3
+ Cs )3
(
= 0 ⇒ s = ⎜
⎛ n Ci
⎞
(
⎟
⎝ C ⎠
−1
)1
3
s = −n C C (exponentially decaying solution)
1
i
(−1)1 3
= −1,
1 ± 3j
2
s =
2, 3
n Ci ⎡1 ± 3 j
⎤ (blows up exponentially because sreal >0 ; but also
2 C ⎣
⎦ | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
d ⎡
( )
ε E t
⎢
⎢
⎣
a a
a
− ε
( )
v t
⎛
⎜
b ⎜ b
⎝
−
⎞⎤
E t a ⎟⎥ 0
⎟
⎥
⎠⎦
( )
=
a
⎛
ε
⎜ a
⎝
+ b
ε a dE
⎞
⎟
b ⎠ dt
a
⎛
+ σ +
⎜ a
⎝
σ a ⎞
b E t
⎟ a
b ⎠
( ) =
( )
ε
b v t
σ
b dv
+
b dt
b
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 24 of 27
B. Step Voltage: v t
( )
= V u t
( )
Then
dv
dt
= V δ t (an impulse)
( )
At t=0
⎛
⎜εa +
⎝
εb
εba dE a =
dv
⎞
⎟
b dt
b ⎠ dt
εb Vδ ( )
t
=
b
Integrate from t=0- to t=0+
t 0
=
ε a dE
+ ⎛
⎜εa + b ⎞
⎟
b ⎠ dt
∫
t 0= − ⎝
⎛
⎜εa +
a dt =
⎝
εba ⎞
b ⎠
⎟ Ea
t 0+
=
t 0= −
=
0
+
∫
t 0
= −
ε
b
b
( )
Vδ t dt =
ε
b
b
V
E t( = 0 − ) = 0
a
⎛
ε
⎜ a
⎝
+
εba ⎞
b ⎠
)
=
E t = | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
R C j
a
a
V(cid:108)
+
R b
ω + 1 R C j ω + 1
b
b
R =
a
a
σa A
, R =
b
b
σb A
ε A
C = a
a
a
ε A
, C = b
b
b
6.641, Electromagnetic Fields, Forces, and Motion
Prof. Markus Zahn
Lecture 7
Page 27 of 27 | https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/06ced4886eba1fc1f0eb037b210e5afb_lecture7.pdf |
18.997 Topics in Combinatorial Optimization
March 4, 2004
Lecture 8
Lecturer: Michel X. Goemans
Scribe: Constantine Caramanis
This lecture covers the proof of the Bessy-Thomass´e Theorem, formerly known as the Gallai
Conjecture. Also, we discuss the cyclic stable set polytope, and show that it is totally dual integral
(TDI) (see lecture 5 for more on TDI systems of inequalities).
1
Recap and Definitions
In this section we provide a brief recap of some definitions we saw in the previous lecture. Also we
answer a question that remained unanswered in the previous lecture regarding the polynomiality of
finding a valid ordering given any strongly connected directed graph.
For a strongly connected digraph D = (V, A), with |V | = n, we make the following definitions.
1. Given an enumeration of the vertices, {v1, . . . , vn}, an arc (vi, vj ) ∈ A is called backward if
i > j and forward if i < j.
2. An ordering O, is an equivalence class of enumerations of a graph. The equivalence class is
defined by the equivalence relations
(a) v1, v2, . . . , vn ∼ v2, v3, . . . , vn, v1,
(b) v1, v2, . . . , vn ∼ v2, v1, v3, . . . , vn, if there is no arc between v1 and v2, i.e., (v1, v2), (v2, v1) /∈
A.
3. Given an ordering O, the index with respect to O of a directed cycle C, denoted iO | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/06d9723de3290adf809e3e61e4dadc8a_co_lec8.pdf |
ordering O, the index with respect to O of a directed cycle C, denoted iO(C), is
the number of backward arcs in C. Recall from the last lecture that the index is well defined,
since the index is invariant under the equivalence operations defined above.
4. We say that an ordering O is valid if for any arc (u, v) ∈ A, there exists a cycle C containing
that arc, with index 1: iO (C) = 1. We showed in the last lecture that there always exists a
valid ordering.
5. A cyclic stable set S with respect to a valid ordering O, is such that S is a stable set on
the underlying undirected graph, and also there exists some enumeration {v1, . . . , vn} of the
ordering such that S = {v1, . . . , vk }, where k = |S|.
Last time we proved that any strongly connected digraph has a valid ordering. In fact, given any
such graph, a valid ordering can be found in time polynomial in the size of the graph. Recall that
the proof of the existence theorem showed that the minimizer of
min
O
�
iO(C),
directed cycles C
must be a valid ordering. Given any ordering O, we showed in the proof that in a polynomial number
of steps (essentially, by repeated “local swaps”), if O is not valid, we can obtain a new ordering O1,
reducing the number of arcs for which there are no cycles of index 1 containing them. Therefore we
can find a valid ordering in polynomial time.
8-1
2 The Bessy-Thomass´e Theorem
Recall the statement of the theorem.
Theorem 1 Given a strongly | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/06d9723de3290adf809e3e61e4dadc8a_co_lec8.pdf |
Bessy-Thomass´e Theorem
Recall the statement of the theorem.
Theorem 1 Given a strongly connected digraph D = (V, A), and a valid ordering O, if αO denotes
the size of the largest cardinality cyclic stable set, then
�
αO = min
iO(Ci),
where the cycles {C1, . . . , Cp} cover the vertex set V .
{C1,...,Cp}
The inequality
αO ≤ min
�
iO(Ci),
{C1,...,Cp}
is straightforward (as each vertex of a cycle stable set must be contained in (at least) one directed
cycle and the corresponding entering arc must be backward), so we consider only the proof of the
reverse inequality.
Before we prove this theorem, we make some remarks. It is important to note that the cyclic
stability number, αO , depends on the ordering O chosen. To illustrate this, recall our digraph on five
vertices from last lecture. In Figure 2, we exhibit two different orderings where the cyclic stability
number is different.
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
� �
Figure 1: In figure (a) above, the cyclic stability number equals 2, where as in (b), the cyclic stability
number equals 1.
Computing the stability number of a general graph is known to be N P -hard. One of the corollar-
ies of the Bessy-Thomass´e | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/06d9723de3290adf809e3e61e4dadc8a_co_lec8.pdf |
number of a general graph is known to be N P -hard. One of the corollar-
ies of the Bessy-Thomass´e Theorem is that the cyclic stability number can be computed efficiently.
iO(Ci) in the right hand
This follows because we can can compute the quantity min
side of the theorem above, efficiently. We can do this by formulating a network flow problem that
computes the minimization. To do this, fix an enumeration of the ordering. Attach a cost of 0 to
every forward arc in the digraph under the given enumeration, and a cost of 1 to every backward arc.
Next, split each vertex v into a pair {vout, v
in) with flow capacity
bounded from below by 1. Then for every arc (u, v) in the original graph, draw an arc (uout, vin) in
the network flow graph. Finding a minimum cost flow in this network can be done efficiently, and
it amounts to finding a set of cycles {C1, . . . , Cp} that cover V , and minimize
} with a directed edge (vout, v
{C1,...,Cp}
�
�
iO (Ci).
in
{C1,...,Cp}
A key step in the proof of the Bessy-Thomass´e Theorem is a lemma that provides a sufficient
condition for a subset S of vertices to be a cyclic stable set.
8-2
Lemma 2 Given a valid ordering O, fix an enumeration, {v1, . . . , vn}. Let S ⊆ V | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/06d9723de3290adf809e3e61e4dadc8a_co_lec8.pdf |
ordering O, fix an enumeration, {v1, . . . , vn}. Let S ⊆ V be a subset of
the vertices. If there are no forward paths between any two vertices of S, then S is a cyclic stable
set.
Suppose, to the contrary, that S has no forward arcs, but S is not a cyclic stable. Let vi
Proof:
be the first element of the enumeration in S. If we rotate the enumeration so that vi becomes v1,
no forward paths are either created or destroyed in S, so we may assume, without loss of generality,
that v1 ∈ S. If S is a cyclic stable set with respect to O, then there exists some enumeration of O
for which the elements of S are the first k = |S| elements of the enumeration. Equivalently, there
exists a sequence of local steps, or swaps we can make according to the equivalence relations defining
an ordering, to move from the current enumeration to one of the correct form. If S is not a cyclic
stable set, as we assume, then this is not possible. Consider the enumeration which brings S “as
close as possible” to having all its elements at the beginning of the enumeration, as illustrated in
Figure 2. By this we mean that as many elements of S as possible are listed first in the enumeration,
and furthermore, the first element of S not part of the initial string of elements of S (which we call
S<) is as close to S< as possible. We denote by S< the elements of S that are at the beginning | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/06d9723de3290adf809e3e61e4dadc8a_co_lec8.pdf |
S< as possible. We denote by S< the elements of S that are at the beginning of
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Figure 2: The figure exhibits the enumeration with respect to which as many elements of S as possible are
the first elements of the enumeration. Since S is assumed not to be a cyclic stable set, there must be some
element w sandwiched by elements of S.
the enumeration, by S> the remaining elements of S, and by W the elements after the last element
of S< and before the first element of S>, as illustrated in Figure 2. Since there are no forward
paths joining any two elements of S, for any w ∈ W there cannot be a forward path from S< to
w, and a forward path from w to an element of S>. Consider the first w ∈ W where there is no
forward path from S< to w (if there is such a vertex). Because w is assumed to be the first such
vertex, there can be no forward path from any vertex v coming before w in the enumeration. If
there were such a vertex v, then if there were a forward path from S< to v, we would also have a
forward path from S< to w. If there | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/06d9723de3290adf809e3e61e4dadc8a_co_lec8.pdf |
to v, we would also have a
forward path from S< to w. If there were no forward path from S< to v, it would contradict our
assumption that w is the first vertex in W that has no forward path from S<. In particular, then,
there are no arcs from any vertex before w in the enumeration, to w. However, there also can be
no arc from w to any vertex before it in the enumeration. This follows because O was assumed to
be a valid ordering. If there were such an arc, say (w, v) for v earlier in the enumeration, because
we assume there are no forward arcs from any vertex coming before w, to w, and that w is the first
such vertex, then any cycle C containing the arc (w, v) must have iO(C) ≥ 2, a contradiction to the
validity of the ordering O. Therefore there are no arcs between w and any vertex previous to w in
the enumeration. But then using the equivalence relations, we can swap w with each element before
it, including then each element of S<. But this contradicts our assumption that the first element of
8-3
S \ S< was as close as possible to S<. Therefore there are no elements in W that have no forward
paths from S. In particular, this implies that there are no forward paths from any w ∈ W to S>.
Then, let vj ∈ S be the first vertex in S>. By assumption, unless W is empty, vj−1 ∈ W , and there
is no forward path from vj−1 to S>, and in particular, (vj−1, | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/06d9723de3290adf809e3e61e4dadc8a_co_lec8.pdf |
∈ W , and there
is no forward path from vj−1 to S>, and in particular, (vj−1, vj ) /∈ A. But then, since O is a valid
ordering, (vj , vj−1) /∈ A. In this case, we can swap the two vertices, contradicting our assumption
that our enumeration put S “as close as possible” to having its elements at the beginning of the
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enumeration. Therefore W must be empty, and S is indeed a cyclic stable set.
We now move to the proof of the Bessy-Thomass´e Theorem.
Proof:
The Main Idea: We want to show that the size of the maximum cyclic stable set equals the minimum
total index of a family of cycles covering V . Essentially the proof relies on mapping our digraph D
to a poset T . At this point, we appeal to Dilworth’s Theorem (lecture 6). Recall that the strong
version of Dilworth’s Theorem tells us that the size of the largest antichain in the poset equals
the minimum number of chains needed to partition the elements of the poset. We show that our
maximum size cyclic stable set S in D, corresponds naturally to an antichain in the poset T . Thus
the size of the largest antichain in T is at least the size of S, i.e., αO. Then we use Lemma 2 to
show that any antichain in T corresponds to a cyclic stable set in D. Thus we have that the size of
the largest antichain in T is exactly αO .
Dilworth’s Theorem now links the number of chains partitioning T to αO . The final | https://ocw.mit.edu/courses/18-997-topics-in-combinatorial-optimization-spring-2004/06d9723de3290adf809e3e61e4dadc8a_co_lec8.pdf |
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