I am interested in compactification of the moduli space of elliptic curves, and I heard that Log geometry is very important for the problem. I am developping the same technique for quantum geometry.

My question was that:

1-Why Log structure can give us a better way to understand degeneration of Elliptic curves? What's the motivation behind?

Is that right that log structure gives us a way to embed the scheme locally into affine space, and the degeneration happens in the space? Like the log structure on $N^2\rightarrow k[x,y]/x.y=0.$, which is fibered over $N\rightarrow k$. In that case, the fourier transform maps $N^2$ into $A^2$, which is the embeding of the nodal curve.

2-Does it make sense to define log-group, i.e. a group with a log structure.

3-Does it make sense to define log structure on a stack which is not algebraic?

Reference for Log geometry: http://www-personal.umich.edu/~satriano/logcurves.pdf

  • $\begingroup$ Maybe you should explain what $N$ and $A$ are? $\endgroup$ – Daniel Larsson Jan 20 '11 at 9:38
  • $\begingroup$ N is just the natural numbers, and A is affine line. It mean $k[N^2]\cong k[x,y]\cong O(A^2)$. It's monoid analog of the Pontryagin duality between Z and $S^1$ $\endgroup$ – Hanh Duc Do Jan 20 '11 at 10:11
  1. Read the introduction in the Kato-Usui book. It's got some nice motivating examples, including a degenerating family of elliptic curves with pictures.

  2. There is no problem with defining a group object in the category of log schemes.

  3. This depends on what you want to do with such a structure, i.e., what properties you need. Maybe you should familiarize yourself with the algebraic case first (e.g., in Olsson's paper Logarithmic geometry and algebraic stacks).

Finally, Matt's notes are a nice introduction, but I hesitate to call them a reference. You might want to look at Ogus's book or Kato's original paper.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.