User rghthndsd - MathOverflow

Dualizing sheaf in mixed characteristic for regular schemes.

2013-03-05T00:02:06Z

I've been looking many places, but everything I find seems to either talk about (a) varieties or (b) extremely general situations with dualizing complexes. As I am not in the situation of (a) (i.e. over a field), and I don't yet understand (b), I was wondering if there was some place which talks about the middle ground. Specifically:

Let $X$ be a Noetherian, regular scheme of finite type over $\mathbb{Z}_p$, but not smooth. Then I've read in many places (even with just Cohen-Macaulay) that the dualizing complex is just a sheaf. Is there an "elementary" description of this sheaf?

Any references would be greatly appreciated.

How to connect monoidal fans (Kato) to fans (Oda).

2013-01-22T14:54:59Z

In the paper Toric Singularities, Kato defines log regular for a sheaf of monoids on a scheme. In section 9, he defines a fan in terms of monoidal spaces (and later that a log regular structure induces such a fan), and in 9.5 connects his concept of a fan to that given by Oda:

Let $L$ be a finitely generated free abelain group. A fan in $L$ in the sense of Oda is equivalent to a fan $F$ given by Kato endowed with a homomorphism of sheaves $h : \text{Hom}(L,\mathbb{Z}) \rightarrow M_F^\text{gp}$ which satisfies three conditions. The last condition is that the map $\text{Mor}(\text{Spec}(\mathbb{N}),F) \rightarrow L$ induced by $h$ is injective (here $\mathbb{N}$ is a monoid under addition).

How does $h$ induce such a map?

Answer by rghthndsd for Basic question about Picard functor

2012-03-10T03:58:18Z

They mean (1) (sort of).

There is a canonical choice of the rigidification $\phi$, so being rigidified doesn't mean there is some map which takes it to the structure sheaf, it is predetermined. Explicitly, the rigidification along the zero section means that $(\varepsilon \circ f, 1_T)^*(L) \cong \mathcal{O}_T$, where $f : T \rightarrow S$ defines $X \times_S T$.

See Mumford's Geometric Invariant Theory, section (d) of 0.5 (pg. 22) for a reference.

Level n-structure as defined by Mumford in GIT

2012-03-09T16:38:23Z

In Mumford's GIT, the definition of level $n$ structure ($n \geq 2)$ is $2g$ sections ${\sigma_1, \dots, \sigma_{2g}} : S \rightarrow A$ such that two conditions hold: (i) For geometric points the sections form a basis for the $n$-torsion points and (ii) $n_A \circ \sigma_i= \varepsilon$.

My question is: Doesn't (i) imply (ii)? Corollary 6.2 implies that since the two maps are equal on geometric fibers by (i), they must be equal everywhere.

Hilbert polynomial of an abelian scheme

2012-02-25T16:09:37Z

This is coming out of Mumford's GIT, section 7.2, page 131.

$A/S$ an abelian scheme of dimension $g$ with polarization $\bar{\omega}$ of degree $d^2$. Then $\pi_*(L^\Delta(\bar{\omega})^3)$ is locally free on $S$ of rank $6^gd$ which defines the closed immersion $\varphi_3 : A \rightarrow \mathbb{P}(\pi_{*}(L^\Delta(\bar{\omega})^3))$. Equip this with a linear rigidification $\phi : \mathbb{P}(\pi_{*}(L^\Delta(\bar{\omega})^3)) \rightarrow \mathbb{P_m} \times S$ so that we get an embedding $I : A \rightarrow \mathbb{P_m} \times S$.

Mumford then states the Hilbert Polynomial of $I(A)$ is easily computed to be $P(X) = 6^gdX^g$.

Exactly how does one go about finding this Hilbert polynomial?

Verifying claims in the proof of the Rigidity Lemma (Mumford, GIT)

2012-01-18T23:35:50Z

In Chapter 6 of Mumford's Geometric invariant theory, during the proof of the rigidity lemma, there are two statements I'm not sure how to verify. The general setup is:

$p : X \rightarrow S$ is flat, $S$ connected, and $H^0(X_s, o_{X_s}) \cong k(s)$ for all points $s \in S$.

In the first part, we're assuming $\epsilon : S \rightarrow X$ is a section, and that $S$ consists of one point. Mumford says: "One checks that $p_*(o_X) \cong o_S$." I found a proof when $p$ is projective (and even proper, I think), which works because this is going to be used on projective abelian schemes, but the general case is still bothering me.
In the second part, $X$ still has the section $\epsilon$, but $S$ is now general (i.e. not just a point), and $p$ is a closed map. During the proof, $Z$ is a closed subscheme of $X$. Mumford claims the statement:

If $p^{-1}(t) \subset Z$ (set-theoretically), for any $t \in S$, then for all artin subschemes $T \subset S$ concentrated at $t$, $Z$ contains $p^{-1}(T)$ as a subscheme.

implies that $Z$ contains an open neighborhood of $p^{-1}(t)$. Intuitively, I think of the artin subscheme as a thickening of the point, and so if I contain an entire fiber then I get "a little bit extra", making $Z$ contain an open neighborhood. I'm wondering how I should do this more formally.

Thanks for any help, it is much appreciated!

References to SGA 8 and descent theory

2012-01-07T16:13:43Z

In Geometric Invariant Theory, by Mumford, Fogarty, and Kirwan, if there is a mention of descent theory, it almost always comes along with a reference to SGA 8, Theorem 5.2 (see the end of the proof of prop 6.9 on page 119, for example). Now I know SGA 8 was never made, but I was wondering:

Does anyone have a good guess as to what this theorem should say?
Does anyone have a good reference for a quick and "hands off" introduction to descent theory? I am really just looking to understand the "gist" of it.

Properties of morphisms induced by divisors on curves

2011-10-10T13:07:55Z

There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim |D| = 1$, and $f$ the induced morphism to $\mathbb{P}^1$.

The degree of $D$ is the degree of $f$, where $\deg(f) = [K(X) : K(\mathbb{P}^1)]$.
In lemma IV.4.2, Hartshorne seems to claim that all points in the support of $D$ are in the same fiber.
Again in lemma IV.4.2, Hartshorne seems to claim that if $[K(X) : K(\mathbb{P}^1)]$ is Galois, then an automorphism of the Galois group permutes elements of the fiber.

Comment by rghthndsd

2013-06-16T22:33:52Z

Is there a good discussion of whether or not the empty scheme is connected somewhere?

Comment by rghthndsd

2013-06-13T00:22:46Z

But where are you mapping that point to? Y must be affine, right?

Comment by rghthndsd

2013-06-08T17:26:56Z

Isn't it immediate that the digits of pi are not random? We can write a finite program to compute them. Or am I missing something?

Comment by rghthndsd

2013-05-23T12:13:41Z

Doesn't looking modulo 3 show that $a$ has to be odd? The equation is $-2^a \equiv p_0^2 (\text{mod } 3)$ and assuming $p_0 \neq 3$ we have $-2^a \equiv 1 (\text{mod } 3)$. Any odd $a$ would satisfy this.

Comment by rghthndsd

2013-03-10T21:57:23Z

I forgot to include: if you have a reference without using dualizing complex, this would be preferred, but not required.

Comment by rghthndsd

2013-03-10T17:32:49Z

The specific properties like local/global duality would be more helpful to me. The $X$ I have in mind is quasi-projective.

Comment by rghthndsd

2013-03-05T14:40:23Z

Thanks, this is exactly what I was looking for. Do you have a reference for this?

Comment by rghthndsd

2013-02-18T22:00:02Z

I haven't checked the notes, but an etale morphism should induce a map $i : k(x) \rightarrow k(y)$. The degree (I believe) should be $[k(y) : i(k(x))]$. This need not be one (as the map $\mathbb{A}^1 \rightarrow \mathbb{A}^1$ sending $x \rightarrow x^n$ shows).

Comment by rghthndsd

2012-12-31T02:27:10Z

"which tells me that $x^4-10x^2+1$ is the minimal polynomial" No, it doesn't. It tells you the minimal polynomial divides this one. Now you have to argue that it's irreducible over $\mathbb{Q}(\sqrt{3})$ (Hint: It isn't).

Comment by rghthndsd

2012-02-25T18:47:51Z

$L^\Delta(\bar{\omega}) = \Delta^*((1_A \times \bar{\omega})^*(P))$, $P$ the Poincare bundle, $\Delta$ the diagonal.

Comment by rghthndsd

2012-01-07T17:47:05Z

Thanks a-fortiori!

Comment by rghthndsd

2011-10-11T17:38:52Z

Ah this makes so much more sense! I was not aware of the "definition" you mentioned; if it's in Hartshorne then I missed it. So for the third, the action of a Galois element fixes the elements of $\mathbb{P}^1$, which I'm identifying with effective divisors linearly equivalent to $D$. Since these divisors also define the fibers of my map, it must be that the fibers are permuted.