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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Deformmation is just an attempt, I cannot promise it will be useful. By using this site, you agree to the Terms of Use and Privacy Policy. The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties. I would appreciate if someone writes an answer either stating 1 Why to study deformation theory?

The so-called Deligne conjecture arising in the context of algebras and Hochschild cohomology stimulated much interest in deformation theory in relation to string theory roughly speaking, to formalise the idea that a string theory can be regarded as a deformation of a point-particle theory. Home Questions Tags Users Unanswered.

Maxim Kontsevich is among those who have offered a generally accepted proof of this. In general, since we harthsorne to consider arbitrary order Taylor expansions in any number of variables, we will consider the category of all local artin algebras over a field. There was also hratshorne MSRI workshop some years ago; I think the videos are still online and there is a draft of a book written by the organizers floating around the web. The existence and the properties of deformations of C require arguments from deformation theory and a reduction to positive characteristic.

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I guess in the process of understanding I will come up with more questions. What follows is an attempt to motivate this beautiful and difficult in my opinion subject. Still many things are vague to me. For our monomial, suppose we want to write out the second order expansion, then. And by thekry way there is another error on the same page, line -1, there is a -2 that should be a So the upshot is: Well, pretend you are a point on a sphere, then to “deform yourself” you have to look around you in all possible directions and see what surrounds you hteory but you need to do this infinitesimally, first because you are a point, and second because deformation theory is the infinitesimal study hartshonre geometric objects.

Everything is done in a special case and shown to follow from basic algebra. Sign up or log in Sign up using Google. Zima 3 A glimpse on Deformation theory by Brian Osserman 4 Robin Hartshorne’s thsory on Deformation Theory Nothing helped me to understand what is deformation theory actually.

For genus 1 the dimension is the Hodge number h 1,0 which is therefore 1. The rough idea is to start with some curve C through a chosen point and keep deforming it until it breaks into several components. In the case of Riemann surfacesone can explain that the complex structure on the Riemann sphere is isolated no moduli.

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

Views Read Edit View history. I think the workshop you mentioned is the following one: It’s not in the link I gave above. So after several repetitions of the procedure, eventually we’ll obtain a curve of genus 0, i. 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: In general you have this: This page was last edited on 31 Octoberat Now you can already see the relation to moduli: Some characteristic phenomena are: For example, in the hheory of hartshore a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit of thoery group action around a given solution.

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

Email Required, but never shown. 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. Infinitesimals have long been in use by mathematicians for non-rigorous arguments in calculus. I am just writing deformatoin comment as an answer. It allows us to answer the question: Sign up using Email and Password.

Robin Hartshorne 2. Algebraic geometry Differential algebra. As it is explained very well in Hartshorne’s book, deformation theory is:. Another method for formalizing deformation theory is using functors on the category of local Artin algebras over a field.

Why on earth should we care about fat points?

## Deformation theory

Post as a guest Name. I am not accepting the answer yet as someone might come up with a more illuminating answer. I understand what is meant by Moduli Space. Dori Bejleri 3, 1 11 If we want to consider an infinitesimal deformation of this space, then we could write down a Cartesian square.

I have tried reading few lecture notes, for example: