User charles siegel - MathOverflow

Langlands Dual Groups

2009-10-22T23:03:48Z

Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does one go about writing down what group it is?

Are quotients of stacks flat?

2012-09-19T08:39:53Z

Let $\cal X$ be a DM stack of finite type over a field (if necessary, I will assume that $k=\mathbb{C}$ and $\cal X$ is a scheme, or even a variety) and $G$ be a finite group. Then we have a quotient stack $[\mathcal{X}/G]$ and in fact we have a natural morphism $\phi:\mathcal{X}\to [\mathcal{X}/G]$.

Is $\phi$ always flat? If not, under what conditions is $\phi$ flat?

Higher Dimensional Gromov-Witten Theories

2009-12-19T23:41:39Z

So, a basic set-up in modern enumerative geometry is that you have some object $X$ (say, a "nice" stack, for whatever definition of "nice" you need) and then you want to "count" the curve of genus $g$ intersecting a bunch of cohomology classes and of degree $\beta$, so you look at $\bar{\mathcal{M}}_{g,n}(X,\beta)$, then pull back the classes, intersect, and get a Gromov-witten Invariant. Famously, this gives the Kontsevich formula counting rational curves in the projective plane passing through $3d-1$ points. And though GW-invariants can be negative and rational, there are nice cases where they do count something legitimate, such as the genus $0$, $n\geq 3$ case into a homogeneous space.

So, enough background, here's my question (and this is largely idle curiosity, so no specific motivation): can we do this in higher dimensions? For instance, given a (smooth?) variety $V$ and marking a bunch of subvarieties $W_1,\ldots,W_n$ (maybe restricting them to points?) can we form $\bar{\mathcal{M}}_{V,(W_1,\ldots,W_n)}(X,\beta)$ a moduli space of stable mappings of varieties deformation equivalent to the one we started with into our space, represented by a given cohomology class $\beta$ and with $W_1,\ldots,W_n$ intersecting some cohomology classes, so that we can get something that can be called higher dimensional Gromov-Witten invariants? If this has been studied, under what conditions does it actually count subvarieties? For instance, if $X$ is $\mathbb{P}^N$ and $V=\mathbb{P}^2$, and maybe if we loosen things to just needing rational maps, could we use something like this to count rational surfaces, satisfying some incidence conditions?

Nonprojective Surface

2009-11-01T00:04:11Z

Let k be an algebraically closed field. It's well known that every complete curve, period, is projective. Also, that every smooth surface is, and that there are smooth 3-folds which are not, and people go to reasonable lengths to include these examples all over the place, so they're easy to find. However, Hartshorne does say that singular complete surfaces are not all projective. Is there a simple example? A complete normal surface that is not projective? Is there some "least singular" possible such surface? I suspect that normality is too much to hope for, but I can't quite phrase why I think this, so is every normal complete surface projective?

Math History books

2010-05-05T19:34:00Z

I'm teaching a course over the summer (it's a sort of make-your-own course for non-majors) and I'm planning on organizing it as a math history course, hitting on major threads through about 1900, and focusing on the evolution of ideas and on people, rather than on the details of proofs. I've also been having a lot of trouble finding a good book covering this material (none finding books on ancient mathematics, but I want to focus on Renaissance to 19th Century, if possible), and so, here's my question:

What would be a good textbook for a course of this nature? Specifically, for a math history course targeted at non-science majors.

When can a finite map be blown up to a flat one?

2011-08-03T18:09:01Z

Let $f:X\to Y$ be a generically finite proper morphism of varieties. There is some locus in $Y$ over which the fiber of $f$ is positive dimensional, so we blow it up, along with the preimage of it in $X$ to get a map $\tilde{f}:\tilde{X}\to\tilde{Y}$ which has finite fibers.

Are there any nice conditions that will guarantee that the map $\tilde{f}$ is flat?

Projectivized Normal Cone to Satake Compactification

2011-05-06T14:07:33Z

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties over $\mathbb{C}$.

There exists a compactification, the Satake compactification, which is minimal and has the property that $$\overline{\mathcal{A}}_g=\mathcal{A}_g\coprod\overline{\mathcal{A}}_{g-1}.$$

It's well known that for a space in $\mathcal{A}_{g-1}$, the projectivized normal cone of the boundary in the whole thing is the Kummer of the point.

What about the higher codimension strata? For instance, what is the projectivized normal cone at a point for the embedding $\overline{\mathcal{A}}_1\subset \overline{\mathcal{A}}_g$, or $\overline{\mathcal{A}}_2\subset \overline{\mathcal{A}}_g$?

Is there a good general method for computing these?

Pullback along the Torelli map is an isomorphism

2011-04-15T22:54:27Z

I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard groups. On the level of rational picard groups, they're both generated by the determinant of the Hodge bundle, but why is this true on the nose? I'll be more than happy with a proof or with a reference to a detailed proof (actually slightly prefer the latter).

How seriously should a graduate student take teaching evaluations?

2010-01-11T21:51:02Z

Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For instance, bad reviews when TAing, but good reviews when actually teaching/lecturing a summer course).

Edit: I asked this question with the situation of a student hoping for a career at research universities in mind, however, I am also interested in other cases.

Edit: In your answer, please mention what your background is: have you served on hiring committees? Are you reporting just what you've heard? Were you successful/unsuccessful in a job search and were told that your teaching evals did/did not make a difference?

Answer by Charles Siegel for Fundamental Examples

2009-11-11T14:14:04Z

The integral $\int \frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$ for $\lambda\neq 0,1$ essentially launched both complex analysis and algebraic geometry, via Riemann's discovery of the Riemann surface that is the natural domain of a function, leading to both analytic theory of Riemann surfaces and to the study of algebraic curves, leading to...complex analysis including in several variables and complex algebraic geometry, as we now know it.

Ramification formula for orbifolds

2011-03-26T15:31:28Z

It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been told that this formula is true for a finite morphism of smooth varieties over $\mathbb{C}$, at least.

First, is the above true?

Second, how must the formula be corrected if $X$ and $Y$ are both smooth orbifolds/DM-stacks?

In particular, what if I want to use that $X$ and $Y$ are of the forms $\tilde{X}/G$ and $\tilde{Y}/H$ for groups $G,H$, and express the formula in terms of objects on $\tilde{X},\tilde{Y}$?

Degrees of compactifications of affine space

2011-03-15T19:24:12Z

Let $k$ be a field and $V\subset \mathbb{P}^n$ a smooth variety over $k$. (Note: not assuming $k$ is algebraically closed). Now assume that for some point (I'm willing to assume every point), the variety $V\setminus T_pV$ is isomorphic to an affine space. What can be said about the degree of $V$?

I know that for $d=2$, this is true. Can it happen for anything that isn't a quadric, or does this property determine that you have a quadric, automatically?

How can I write down polynomial relations that define when a polynomial is a square?

2011-02-11T20:02:46Z

It's easy to tell when a polynomial is squarefree (or not): that's just the question of the vanishing of the discriminant, which can be dealt with as the resultant of $f$ and $f'$. However, given a polynomial of degree $2n$ $f$, when is it of the form $g^2$ for $g$ a polynomial of degree $n$?

I've been trying to work out the relations on the coefficients that will guarantee this for a specific degree ($n=6$ is my case) but whenever I take the obvious equations in the coefficients of $g$ and of $f$ and try to use Groebner bases to eliminate the coefficients of $g$, I run out of memory and my software crashes. Is there a way to understand the locus of polynomials which are squares concretely without having to do a (seemingly unrealistically) big computation? Or perhaps a clever trick that will give these polynomial identities in a more computable way?

Answer by Charles Siegel for New proofs to major theorems leading to new insights and results?

2011-02-11T12:29:21Z

If you're interested in something that's expected to do this (here, with a related paper here), but is a very current project, there's Lazic's proof of the finite generation of (log) canonical rings. I don't know of any great insights gained from the new proof, other than the surprising fact that it's POSSIBLE to prove it this way, and that the method, rather than requiring the Mori program to prove the theorem, allows a proof of many important theorems in the Mori program from it. This is, though, quite a work in progress.

Answer by Charles Siegel for Video lectures of mathematics courses available online for free

2011-02-05T22:51:03Z

Ted Chinburg has videos of his lectures for what is going on a 2 year course in algebraic number theory online( direct links to videos: semester 1, semester 2, semester 3, semester 4), and from there you can also get lectures from various seminars at Penn.

Also, there's the MSRI database for all the things that go on there, they're all over the website at each program's site.

Different ways to construct maps and the tensor products of line bundles

2011-02-04T14:35:42Z

Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms from $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective space, which may or may not be injective, so you get a map to another curve. To get a morphism to $C$, a particular case is to take a line bundle $\mu$ of order 2, which gives us a curve $\tilde{C}$ as a double cover of $C$.

How do these two methods interact? For instance, given a curve $C$, and the tower $\tilde{C}\to C\to \bar{C}$ with the first map given as a double cover by $\mu$, and the second map given by a line bundle $L$, presumably the geometry of this tower is connected to the line bundle $L\otimes\mu$, but it's not obvious to me how to connect the two notions to get any actual information.

Answer by Charles Siegel for Modular forms of fractional weight

2011-01-24T02:48:27Z

Modular forms of weight 1/2 are actually quite prominent in geoemtry (I can't speak for number theory). For instance, the 2nd order theta functions (which encode information about points of order two on abelian varieties, for instance) are of weight 1/2. They give a natural and important map from a certain cover of the moduli space of abelian varieties (specifically $\mathcal{A}_g^{(2n,4n)}$) into projective space which is injective for $n\geq 2$. Here are a few reference for 2nd order theta functions:

Kummer varieties and the moduli spaces of abelian varieties - van Geemen and van der Geer

Igusa's book on Theta Functions

Mumford's Tata Lectures on Theta.

Grushevsky's survey of the Schottky Problem (lots of things on the Schottky problem involve 2nd order theta functions)

Answer by Charles Siegel for What would one call this generalisation of the moduli space of theta-characteristics

2010-12-30T20:49:53Z

I've seen many people referring to what you call $r$-spin Riemann surfaces as higher spin curves (sometimes with the $r$, sometimes suppressed), so my thought would be to call it the moduli stack of $r$-theta characteristics or of higher theta characteristics, if the $r$ isn't as important.

Reference for the Hodge Bundle

2010-11-29T23:51:19Z

For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to $B$ along the zero section of the sheaf of relative differentials. The most interesting examples are when $B$ is $M_g$ or $A_g$, and the fibrations are the Jacobian fibration or the universal family of abelian varieties.

Is there a good reference for the properties of this bundle? And for the determinant line bundle on these spaces? Things like self-intersection numbers, cohomology (in particular global sections) are particularly of interest.

Chern classes of pushforwards

2010-12-18T21:19:17Z

Let $f:X\to Y$ be a proper morphism of normal varieties (smooth as DM stacks, but I only care about the coarse spaces). The map $f$ is generically finite, but not flat (so no hope of smoothness and Grothendieck-Riemann-Roch applying) and I have a fairly detailed understanding of the fiber over any point, images of $f$ restricted to divisors and so forth.

Now, take a divisor $D$, and identify it with an invertible sheaf in the standard way. I'm looking for a way to compute the first Chern class of $f_*D$ on $Y$.

Answer by Charles Siegel for What would you want to see at the Museum of Mathematics?

2010-12-26T17:16:49Z

Some Pixar/Dreamworks stuff might be good...a Pixar guy gave a cool talk at ICM a few years ago about the mathematics they use to do the 3d rendering, topping it with harmonic coordinates.

Canonical bundle of compactifications

2010-12-10T21:20:01Z

Let $X$ be a quasi-projective variety. Suppose that we (perhaps partially, if either enough is known) compactify to $\bar{X}$ with $\bar{X}\setminus X=D$ is a divisor. Say that we know the canonical bundle $K_X$. Then $K_{\bar{X}}=K_X+nD$ for some $n$.

Is $n$ always negative? The examples I'm thinking of are for $X=\mathcal{M}_g,\mathcal{A}_g$
Is there a good method for computing this $n$?

In both cases, I'm particularly interested in finite covers of $\mathcal{M}_g,\mathcal{A}_g$ and other moduli spaces, if that helps to know how the variety is given.

Unstable Vector Bundles

2009-10-21T01:52:47Z

As a follow up to me other question, what can be said about unstable vector bundles? I know this is rather open ended, but what sorts of horrible things does having a subbundle of strictly greater slope imply?

Picard Groups of Moduli Problems

2010-05-20T04:43:20Z

First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible.

I'm told that for $g\geq 2$ it is known that the Picard groups of $\mathcal{M}_g$ and $\mathcal{A}_g$ (the moduli spaces of curves of genus $g$ and abelian varieties of dimension $g$) are both isomorphic to $\mathbb{Z}$ (at least, over $\mathbb{C}$). What's the most efficient way to compute this? In fact, for $\mathcal{M}_g$, it's even generated by the Hodge bundle, I'm told. Ideally I want to avoid using stacks (though if stacks give an elegant proof, I'm open to them) and also would like to be able to calculate the degrees of some natural bundles, though I get that that's going to be a bit harder, so I want to focus this question on the computation of the Picard group.

Algebraic Curves and Phase Diagrams of Physical Systems

2010-11-04T02:51:49Z

Lots of low degree curves arise naturally as the phase spaces of physical systems (that is, the curve parameterized by $(q,p)$ where $q$ is a generalized position variable and $p$ is a generalized momentum variable (such that $p=\dot{q}$, etc).

For instance, if degree is 1, then we can construct the curve as the phase space of a particle moving with constant velocity, and it is parameterized by $(x,0)$, so is given by the equation $y=0$, up to a choice of coordinates.

For $d=2$, a conic, we can use, for instance, a spring, whose position is $\cos(x)$, and so $(\cos x,\sin x)$ parameterizes a circle (ellipses are also fairly easy to see how to do this way, and parabolas are linearly accelerated particles, though I don't see hyperbolas immediately)

For $d=3$, it is well known that a simple pendulum has phase space an elliptic curve with roots the initial height, the length of the pendulum, and minus the length.

Are there other nice examples like this? Is there a natural physical system that realizes hyperbolas? Does every real elliptic curve arise in this way? How about quartic or higher curves? I assume there are physical systems that work, as every ODE is modeling SOME system, but are there well-known examples that arise naturally in physics?

Answer by Charles Siegel for Help with Griffiths & Harris, Surfaces

2010-11-01T13:19:21Z

I don't have my copy of Griffiths and Harris in front of me, but regarding Grothendieck's theorem, there's a nice elementary argument by Michiel Hazewinkel & Clyde F. Martin. Here is a link to ScienceDirect. It's only five pages, and mostly consists of linear algebra (and really, is closer to 3 pages, ignoring abstract, intro, and white space at the end).

Answer by Charles Siegel for How do I stop worrying about root systems and decompostion theorems (for reductive groups)?

2010-10-21T02:21:29Z

For a very non-Lie Theoretic reason to care about the ADE root systems, look at http://front.math.ucdavis.edu/0809.2579 and do exercise 66 on DuVal singularities of surfaces. It's good to do by hand if you need practice with blowups, otherwise, just looking at the fact that you get ADE out of canonical surface singularities should motivate the need for Dynkin diagrams, which describe the root systems.

Answer by Charles Siegel for Spin structures on the Grassmannians

2010-10-12T19:44:38Z

As far as when do spin structures exist, a manifold is spin if and only if the 2nd Stiefel Whitney class $w_2(X)=0$, which is the same as $c_1(X)\mod 2=0$. So we must calculate $c_1(T_X)$.

Let $R$ be the universal subbundle and $Q$ the universal quotient bundle. Then we have $0\to R\to \mathbb{C}^n\to Q\to 0$ for $Gr(k,n)$, and the total Chern classes satisfy $c(R)c(Q)=1$ Thus, $c_1(R)+c_1(Q)=0$, and we can show that $c_1(R)=-1$ and thus $c_1(Q)=1$. But the tangent bundle is $\hom(R,Q)=R^*\otimes Q$, which means that $c_1(T)=n$, and completely ignores $k$.

So in particular, $\mathbb{P}^n=Gr(1,n+1)$ has Chern class $n+1$, and so will be spin if and only if $n$ is odd.

I don't know the answers to 2 and 3.

Note: as Dave pointed out in the comment, I've identified $H^2$ with the integers for Grassmannians because there is a unique Schubert class $\sigma_1$ which is an ample generator, and so we do have a canonical identification. This is trickier for other spaces, of course.

Tangent Space to a Linear System

2010-10-09T02:53:19Z

Let $X$ be a smooth projective variety and let $D$ be an effective divisor on $X$. Is there a natural way to describe the tangent space to $|D|$ (or $|D|^\vee$, of course) at a divisor $D'$? Preferably as some sort of cohomology group, again ideally on $X$. I would prefer to avoid using the fact that the linear system is a projective space, and do things naturally.

Answer by Charles Siegel for Is the normalization map bijective?

2010-10-03T21:42:52Z

No, the map isn't generally bijective. For curves, normalization is the same as resolution of singularities. Look at a nodal cubic curve $y^2=x^2(x-1)$ in the plane. Over the node, the normalization has two points.

Comment by Charles Siegel

2011-08-03T19:07:22Z

Perhaps you should try plugging it into maple or a similar program? Either way, this question isn't appropriate for this website, which is geared towards research level mathematics.

Comment by Charles Siegel

2011-07-09T15:05:18Z

Is there a reason to avoid just using Groebner bases to find equations for the image, rather than finding the global sections of the twisted ideal?

Comment by Charles Siegel

2011-06-29T20:43:57Z

Am I missing something, or is your map actually to the Kummer of the Jacobian? How do you send the set $\{x,y\}$ to $x-y$ without a sign ambiguity?

Comment by Charles Siegel

2011-05-06T16:40:55Z

Thanks, I used to know that, but had forgotten

Comment by Charles Siegel

2011-05-05T01:47:11Z

I believe I made it readable

Comment by Charles Siegel

2011-04-22T23:27:06Z

would you accept applications of affine algebraic geometry to real world problems?

Comment by Charles Siegel

2011-04-18T16:20:06Z

Very helpful! Is it by any chance known to generalize, say, to a finite cover of the moduli spaces (like level n, (n,2n), or the Prym moduli spaces)?

Comment by Charles Siegel

2011-03-26T20:28:29Z

Thanks a lot! Though seeing this answer, I think I'm realizing that the question I asked and the one I meant to ask are slightly different. Though the underlying varieties I have are normal, so all of this goes through, it's actually important for me to keep track of some stacky structure, because my group actions aren't faithful (at the least, I know that my group action on the domain has generic stabilizer nontrivial, and I think the action on the codomain might, but I haven't analyzed it completely yet.)

Comment by Charles Siegel

2011-03-26T20:26:32Z

@Sándor: Yes, the map I have does actually come from a map $\tilde{X}\to \tilde{Y}$.

Comment by Charles Siegel

2011-03-16T15:39:20Z

@quim I was just about to ask if it is true for hypersurfaces, or if it becomes true if I say instead of $V\setminus T_pV$, $V\setminus H$ where $H$ is a hyperplane containing $T_pV$.

Comment by Charles Siegel

2011-02-26T18:50:15Z

@Chulumba, there are tons of answers for your $g$, none of which are interesting, really. Look at $x-y$. Then, for the infinitely many pairs $(x,y)$ where $y=2x$, it will give the gcd, in fact, $x-ay$ works analogously.

Comment by Charles Siegel

2011-02-12T19:46:22Z

@Igor: This is essentially what I was doing in Macaulay2, but the elimination step was too bad by the time I got to 12th degree polynomials

Comment by Charles Siegel

2011-02-12T19:44:54Z

When I said "$n=6$", I meant the case where $\deg f=12$, which makes this much, much worse for a CAS to work out.

Comment by Charles Siegel

2011-02-12T19:43:39Z

@Thierry: I always use Macaulay2, and it broke for the case of a degree 12 polynomial (at least, on my computer) Regarding algebraic closure issues: I was working over $\mathbb{C}$, but it's interesting how this question works for non-closed fields.

Comment by Charles Siegel

2011-02-11T20:47:48Z

That would be fine for determining if a given polynomial is a square, but I'm looking for the equations that define the locus of square polynomials, and I don't see how to do that in this way.