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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Brenin I have now some idea about deformation. Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a Desingularization of moduli varities for vector bundles on curves, Int. Here is MSE copy: Home Questions Tags Users Unanswered.

Thanks for discovering the error. Some characteristic phenomena are: One of the major applications of deformation theory is in arithmetic. For genus 1 the dimension is the Hodge number h 1,0 which is therefore 1. Still many things are vague to me.

I am not accepting the answer yet as someone might come up with a more illuminating answer. Some of the above mentioned notes say that deformation theory is somehow related to Moduli Theory.

Deformation theory

Why on earth should we care about fat points? As it is explained very well in Hartshorne’s book, deformation theory is: Maxim Kontsevich is among those who have offered a generally accepted proof of this. So after several repetitions of the procedure, eventually we’ll obtain a curve of genus 0, i. Now let me tell you something very naive.


Robin Hartshorne 2. 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

May be I am missing some points for understanding. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. This is true for moduli of curves. For genus 1, an elliptic curve has a one-parameter family of complex structures, as shown in elliptic function theory. Home Questions Tags Users Unanswered. I do not have the book in front of me, but it sounds to me like the formulation above is false.

Deformation theory – Wikipedia

Replacing C by one of the components has the effect of decreasing either the genus or the degree of C. I am just writing my comment as an answer. In the case of genus 0 the H 1 vanishes, also.

I have tried reading few lecture notes, for example: Zima 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.

Sign up or log in Sign up using Google. May be as I read more I will understand it better. To motivative the definition of a pre-deformation functor, consider the projective hypersurface over a field.

Good references are online notes by Ravi Vakil, and Sernesi’s book Deformations of algebraic schemes. 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. Ravi’s introductory lecture was probably good, although it has been a long time since I’ve watched it. Spencerafter deformation techniques had received a great deal of more tentative application in the Italian school of algebraic geometry.


Mathematics Stack Exchange works best with JavaScript enabled. Sign up using Email and Password. The phenomena turn out to be rather subtle, though, in the general case. This is now accepted as proved, after some hitches with early announcements.

The general Kodaira—Spencer theory identifies as the key to the deformation theory the sheaf cohomology group. I guess in the process of understanding I will come up with more questions. These obviously depend on two parameters, a and b, whereas the isomorphism classes of such curves have only one parameter. 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.

I’ll tell you later what nice group describes these objects! In general you have this: MathOverflow works best with JavaScript enabled. Views Read Edit View history.