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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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Seshadri and b Cohomology of certain moduli spaces of vector bundles Proc. The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties. I understand what is meant by Moduli Hrtshorne. Brenin 8, 3 15 A pre-deformation functor is defined as a functor. In the case of genus 0 the H 1 vanishes, also. MathOverflow works best with JavaScript enabled.

Spencerafter deformation techniques had received a great deal of more tentative application in the Italian school of algebraic geometry.

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: Brenin I have now some idea about deformation.

The theeory of the proof later became hheory as Mori’s bend-and-break. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. But I have no clue how. In general you have this: We could also interpret deformxtion equation as the first two terms of the Taylor expansion of the monomial.


To motivative the definition of a pre-deformation functor, consider the projective hypersurface over a field. In general, since we want to consider arbitrary order Hartshorbe expansions in any number of variables, we will consider the category of all local artin algebras over a field. Everything is done in a special case and shown to follow from basic algebra.

So the upshot is: What follows is an attempt to motivate this beautiful and difficult in my opinion subject. Hartshorn clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and hagtshorne policyand that your continued use of the website is subject to these policies.

Ravi’s introductory lecture was probably good, although it has been a long time since I’ve watched it. Mathematics Stack Exchange works best with JavaScript enabled. Some of the above mentioned notes say that deformation theory is somehow related to Moduli Theory. There we found another strong link with moduli!

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

Here is MSE copy: The general Kodaira—Spencer theory identifies as the key to the deformation theory the sheaf cohomology group. These examples are the beginning of a theory applying to holomorphic families of complex manifolds, of any dimension. Now let me tell you something very naive. Considering families over a fatter point, e. You are welcome to ask more questions as soon as you read more about it!

Then, the space on the right hand corner is one example of an infinitesimal deformation: Babai 2, 2 15 Since we want to consider all possible expansions, we will let our predeformation functor be defined on objects as. Replacing C by one of the components has the effect of decreasing either the genus or the degree of C.


One might think, in analogy, of a structure that is not completely rigid, and that deforms slightly to accommodate forces applied from the outside; this explains the name. For genus 1 the dimension is the Hodge number h 1,0 which is therefore 1. Algebraic geometry Differential algebra.

Deformation theory – Wikipedia

For example, in the hartsjorne 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.

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.

I do not have the book in front of me, but it sounds to me like the formulation above is false. Post as a guest Name. If we want to consider an infinitesimal deformation of this space, then we could write down a Cartesian square.

Deformation theory

It is just an attempt, I cannot promise theorry will be useful. I’ll tell you later what nice group describes these objects! Still many things are vague to me.

And by the way there is another error on the same page, line -1, there is a -2 that should be a Why on earth should we care about fat points?

The existence and the properties of deformations of C require arguments from deformation theory and a reduction to positive characteristic.