User oldmacdonaldhadaform - MathOverflow

Enumerativity of Gromov-Witten invariants of orbifolds

2013-01-31T11:02:06Z

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf.

Is there some sense, or some class of examples (that include true orbifolds, namely orbifolds that are not schemes), in which these numbers have some enumerative meaning?

One class of examples is Gorenstein orbifolds, namely those orbifolds whose all degree shifting numbers are integers. In this case one can use the crepant resolution conjecture to translate the question (at least in genus $0$) into a question of enumerativity of Gromov-Witten invariants for the crepant resolution.

But what about other cases? In particular, can one make some enumerative sense of the Gromov-Witten invariants that are obtained when at least one of the cohomology classes is chosen to be the cohomology class of a twisted sector?

(apologies if the question is too vague, and if it is a repetition of other similar questions. I am not sure if this should be community wiki.)

A question on the Picard group

2013-01-29T21:03:46Z

Let $X$ be a simply connected smooth projective variety, whose Picard group is generated by the classes of the irreducible codimension 1 loci $D_1, \ldots, D_k$. Let $E_1, \ldots, E_r$ be other irreducible codimension 1 loci, and suppose that $X^0$ is the complement in $X$ of the divisors $D_i$ and $E_j$.

Suppose now that $X_0$ is the complement of $n$ irreducible loci of codimension $1$ in $Y$, a smooth projective variety.

Question: Can I conclude that the Picard group of $Y$ has rank $n-r$?

I can answer the question affirmatively over $\mathbb{C}$, by using the long exact sequence with compact support associated with the inclusion $Y \setminus X^0 \to Y$, but I would like to know if there is an algebraic proof of this (valid over any algebraically closed field $k$).

EDIT: As pointed out in the answer, I am actually assuming that the Picard group of $X$ is FREELY generated by the $D_1, \ldots, D_k$.

Is $M_{1,n}$ affine?

2012-07-17T11:24:05Z

A famous conjecture of Looijenga states that the moduli space of curves $M_{g,n}$ is the union of $g- \delta_{0,n}+ \delta_{0,g}$ open affine subsets, where $g,n$ are non-negative integers satisfying $2g-2+n>0$, and $\delta$ is the Kronecker delta.

I know of proofs of this conjecture in the case $(g,0)$ for $2 \leq g \leq 5$ (Fontanari-Looijenga and Fontanari-Pascolutti), and in the case $(0,n)$ for all $n$'s.

Is this conjecture true for $M_{1,n}$ ? Namely, is it known if $M_{1,n}$ is affine?

(Are there other cases when the conjecture is known?)

Answer by OldMacdonaldHadaForm for Flatness for family of hypersurfaces

2012-05-18T06:34:26Z

Consider the projective space of degree $d$ monomials in $n+1$ variables $\mathbb{P}^{\binom{n+d}{d}-1}$. On this projective space there is a universal family of hypersurfaces in $\mathbb{P}^n$. This family is flat since the Hilbert polynomial is constant, and the base is reduced.

Now given any family $X \to Y$ of not necessarily flat hypersurfaces of degree $d$ in $\mathbb{P}^n$, there exists a unique moduli map $Y \to \mathbb{P}^{\binom{n+d}{d}-1}$ such that $X\to Y$ is the pullback of the universal family via the moduli map. Since flatness is stable under base change, the map $X \to Y$ must also be flat even when $Y$ is not reduced.

NOTE: I suspect that this answer may contain a mistake due to some sort of "circular" reasoning. At the moment I can not see this mistake though.

Flatness for family of hypersurfaces

2012-05-10T22:08:22Z

Let $X \to Y$ be a family of hypersurfaces in a constant $\mathbb{P}^n$, i.e. $X \subset Y \times \mathbb{P}^n$ is locally on $Y$ given by one equation of degree $d$ in $\mathbb{P}^n$.

Is $X \to Y$ automatically flat? I know that it is so if $Y$ is reduced, since in this case the fact that the Hilbert polynomial of $X_y$ is constant on $Y$ implies that the family is actually flat. So is $X \to Y$ still automatically flat when $Y$ is nonreduced?

Intersections with divisors on moduli of curves

2012-05-11T14:40:28Z

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of stable curves of genus $g$ with $n$ points.

Consider

$0 \neq \gamma \in Pic(\overline{\mathcal{M}}_{g,n})$

the first Chern class of a line bundle (note that the Picard coincides with $H^2(\overline{\mathcal{M}}_{g,n}, \mathbb{Z})$ and with the Neron-Severi group in this case)

It is well-known from Arbarello-Cornalba-Harer that the group of first Chern classes is generated by $\kappa_1$ and by $\delta$-classes and $\psi$-classes.

My question: is the multiplication by $\gamma$ an isomorphism:

$ H^{3g-3+n-1} \to H^{3g-3+n+1} \ ? $

(as it happens in the Hard Lefschetz theorem when $\gamma$ is an hyperplane section). Are there $\gamma$s for which it is not an isomorphism?

This is of course true when $\gamma$ is in the ample cone or in the antiample cone.

Modular curve parametrizing two cyclic subgroups of an elliptic curve

2012-04-25T08:31:16Z

The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing an elliptic curve $E$, together with two cyclic subgroups $G$ and $H$, of order respectively $M$ and $N$, of the group of degree $0$ line bundles modulo linear equivalence on $E$.

(note that the subgroup generated by $G$ and $H$ need not be of rank $2$, although it will be so generically)

When $N=1$, $Y_0(M,N)$ is just the well known modular curve $Y_0(M)$.

Questions:

1) Are there references for this moduli space/modular curve? Is this well studied/well known?

2) I think of this as the fiber product $Y_0(M) \times_{\mathcal{M}_{1,1}} Y_0(N)$ (the fiber product over the moduli space of elliptic curves). Is this correct?

3) What can we say about the curve $Y_0(M,N)$? Is it irreducible? Are there some kind of formulas for its genus as there are for $Y_0(M)$?

Comment by OldMacdonaldHadaForm

2013-02-01T08:27:04Z

Thank you very much!

Comment by OldMacdonaldHadaForm

2013-01-31T10:18:25Z

Thanks a lot for your answer. I picked the other one because I could understand better the details ;)

Comment by OldMacdonaldHadaForm

2013-01-31T10:14:34Z

Thank you very much. This is much more natural and elementary than the approach I was following.

Comment by OldMacdonaldHadaForm

2013-01-30T15:09:42Z

Thanks! I think my question is solved affermatively now.

Comment by OldMacdonaldHadaForm

2013-01-30T10:00:41Z

The point is that I am not sure how the theory of (mixed) Hodge structures on the ètale cohomology groups works, and if it exists at all.

Comment by OldMacdonaldHadaForm

2013-01-30T09:59:29Z

What I prove is that H^2(Y)=H^{1,1}(Y) has rank $n-r$: I have an exact sequence $$ 0 \to H^2(Y) \to \mathbb{Q}^n \to H^1(X^0)= \mathbb{Q}^r \to H^1(Y) $$ and I have to proove the surjectivity of $\mathbb{Q}^n \to \mathbb{Q}^r$ to conclude. To do so, I prove $\mathbb{Q}^r \to H^1(Y)$ is the zero map, by using the purity of the Hodge structures.

Comment by OldMacdonaldHadaForm

2012-07-18T07:45:31Z

Thank you very much!

Comment by OldMacdonaldHadaForm

2012-07-17T18:56:12Z

Perhaps there is still something to say about this. Indeed, there is no universal curve over the moduli scheme $M_{g,n}$, which is what we are considering. Do you have a reference?

Comment by OldMacdonaldHadaForm

2012-07-17T13:36:35Z

Wow. Thank you!

Comment by OldMacdonaldHadaForm

2012-07-17T11:51:18Z

Thank you very much! How do you see that the morphism is affine?

Comment by OldMacdonaldHadaForm

2012-05-17T10:33:50Z

Sorry for the previous question. I think I understand now.

Comment by OldMacdonaldHadaForm

2012-05-16T09:00:00Z

Thank you very much for your answer.

Comment by OldMacdonaldHadaForm

2012-05-16T08:59:22Z

This is so very beautifully concrete! May I ask for a reference for the "general" Weierstrass form? I know it only over $\mathbb{C}$ (or in characteristic zero).

Comment by OldMacdonaldHadaForm

2012-05-13T06:54:10Z

May I ask if a similar result holds for the Hilbert scheme of points? Namely, if a family of $0$-dimensional subschemes of $Y \times \mathbb{P}^n$ has constant Hilbert polynomial $k$, is it automatically flat over $Y$?

Comment by OldMacdonaldHadaForm

2012-05-11T18:30:31Z

Dear Jason Starr, thank you very much for your comment. I will rephrase the question to include only the case $k=1$, which is the one I was originarily interested in.