From the reviews: “Robin Hartshorne is the author of a well-known textbook from which several generations of mathematicians have learned modern algebraic. In the fall semester of I gave a course on deformation theory at Berkeley. My goal was to understand completely Grothendieck’s local. I agree. Thanks for discovering the error. And by the way there is another error on the same page, line -1, there is a -2 that should be a

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## Seminar on deformations and moduli spaces in algebraic geometry and applications

Another application of deformation theory is with Galois deformations. Deformxtion 3 A glimpse on Deformation theory by Brian Osserman 4 Robin Hartshorne’s book on Deformation Theory Nothing helped me to understand what is deformation theory actually. Now let me tell you something very naive. This is now accepted as proved, after some hitches with early announcements.

### Seminar on deformations and moduli spaces in algebraic geometry and applications

There is an obstruction in the H 2 of the same sheaf; which is always zero in case of a curve, for general reasons of dimension. Some characteristic phenomena are: In a “neighborhood” of this member of the family, all other curves are smooth conics, so when we stare at this unique, very special singular conic, the natural question arises: And by the way there is another error on the same page, line -1, there is a -2 that should be a Home Questions Tags Users Unanswered.

There we found another strong link with moduli! This is true for moduli of curves. A simple application of this is that we can find the derivatives of monomials using infinitesimals:. Hence there must be an equation relating those a and b which describe isomorphic elliptic curves. Deformation theory was famously applied in birational geometry by Shigefumi Mori to study the existence of rational curves on varieties. I have tried reading few lecture notes, for example: Retrieved from ” https: Post Your Answer Discard By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

A pre-deformation functor is defined as a functor. Spencerafter deformation techniques had received a great deal of more tentative application in the Italian school of algebraic geometry.

Good references are online notes by Ravi Vakil, and Sernesi’s book Deformations of algebraic schemes. Now you can already see the relation to moduli: The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties.

Since we are considering the tangent space of a point of some moduli space, we can define the tangent space of our pre -deformation functor as. I think the workshop you hqrtshorne is the following one: It is typically the case that it is easier to described the functor for a moduli problem instead of finding an actual space.

For example, in the geometry of numbers a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit of a group action around a given solution. Thank you for your elaborate answer.

If we have a Galois representation. The phenomena turn out to be rather subtle, though, in the general case. Email Required, but never shown. So it turns out that to deform yourself means to choose a tangent direction on the sphere. The general Kodaira—Spencer theory identifies as the key to the deformation theory the sheaf cohomology group. May be as I read more I will understand it better.

Infinitesimals can be made rigorous using nilpotent elements in local artin algebras. Why on earth should we care about fat points?

I would appreciate if someone writes an answer either stating 1 Why to study deformation theory? These are very different from the first order one, e.

I am not accepting the answer yet as someone might come up with a more illuminating answer. Sign up or log in Sign up using Google. Brenin 8, 3 15 Algebraic geometry Differential algebra.

But I have no clue how. One of the major applications of deformation theory is in arithmetic.

## Deformation theory

You are welcome to ask more questions as soon as you read more about it! For genus 1, an elliptic curve has a one-parameter family of complex structures, as shown in elliptic function theory. Post Your Answer Discard By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

In general, since we want to consider arbitrary order Taylor expansions in any number of variables, we will consider the category of all local artin algebras over a field. All the lectures seems to be very abstract to me. I do not have the book in front of me, but it sounds to me like the formulation above is false. The existence and the properties of deformations of C require arguments from deformation theory and a reduction to positive characteristic.