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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May be I am missing some points for understanding. Since we want to consider all possible expansions, we will let our predeformation functor be defined on objects as. Brenin 8, 3 15 Martin Sleziak 2, 3 20 I’ll tell you later what nice group describes these objects!

If we want to consider an infinitesimal deformation of this space, then we could write down a Cartesian square. So the upshot is: So it turns out that to deform yourself means to choose a tangent direction on the sphere.

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

I would appreciate if hartsborne writes an answer either stating 1 Why to study deformation theory? Sign up or log in Sign up using Google. Sign up or log in Sign up using Google. Some characteristic phenomena are: Now let me tell you something very naive.

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The existence and the properties of deformations of C require arguments from deformation theory and a reduction to positive characteristic. Home Questions Tags Users Unanswered. Dori Bejleri 3, 1 11 Now you can already see the relation to moduli: Email Required, but never shown.

For genus 1 the dimension is the Hodge number h 1,0 which is therefore 1. One of the major applications of deformation theory is in arithmetic. I understand what is meant by Moduli Space. Hattshorne you give any link for that “draft”? In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. Thank you for your elaborate answer. There hatrshorne also an 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.

Why on earth should we care about fat points? Sign up using Email and Password. Some of the above mentioned notes say that deformation theory is somehow related to Moduli Theory. Perturbation theory also looks at deformations, in general of operators.

These examples are the beginning of a theory applying to holomorphic families of complex manifolds, of any dimension. We could also interpret this equation as the first two terms of the Taylor expansion of the monomial.

I am just writing my comment as an answer. I came across these words while studying these papers a Desingularization of moduli varities for vector bundles on curves, Int. Post Your Answer Discard By clicking “Post Your Answer”, you acknowledge that haetshorne have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

Hence there must be an equation relating those a and b which describe isomorphic elliptic curves. The rough idea is to start with some deformarion C through a chosen point and keep deforming it until it breaks into several components.

I do not have the book in front of me, but it sounds to me like the formulation above is false. 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. As it is explained very well in Hartshorne’s book, deformation theory is:.

### algebraic geometry – Studying Deformation Theory of Schemes – Mathematics Stack Exchange

There we found another strong link with moduli! That formulation is false. For genus 1, an elliptic curve has a one-parameter family of complex structures, as shown in elliptic function theory.

I am finding it difficult to understand why everyone suddenly starts talking about artinian local algebras. Sign up using Facebook. Here is MSE copy: Robin Hartshorne 2. Algebraic geometry Differential algebra. Another method for formalizing deformation theory is using functors on the category of local Artin algebras over a field. Babai 2, 2 15 This was put on a firm basis by foundational work of Kunihiko Kodaira and Donald C. 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.

Post as a guest Name. Good references are online notes by Ravi Vakil, and Sernesi’s book Deformations of algebraic schemes.