By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I came across these words while studying these papers a Desingularization of moduli varities for vector bundles on curves, Int. Symp on Algebraic Geometry by C. Seshadri and b Cohomology of certain moduli spaces of vector bundles Proc.

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By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I came across these words while studying these papers a Desingularization of moduli varities for vector bundles on curves, Int. Symp on Algebraic Geometry by C.

Seshadri and b Cohomology of certain moduli spaces of vector bundles Proc. Indian Acad. Doubek, M. Markl and P. I am finding it difficult to understand why everyone suddenly starts talking about artinian local algebras. All the lectures seems to be very abstract to me.

I would appreciate if someone writes an answer either stating 1 Why to study deformation theory? I understand what is meant by Moduli Space. Some of the above mentioned notes say that deformation theory is somehow related to Moduli Theory. But I have no clue how. What follows is an attempt to motivate this beautiful and difficult in my opinion subject. It is just an attempt, I cannot promise it will be useful. As it is explained very well in Hartshorne's book, deformation theory is:. Now you can already see the relation to moduli: we just finished talking about a "family of curves" Now let me tell you something very naive.

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 - but you need to do this infinitesimally, first because you are a point, and second because deformation theory is the infinitesimal study of geometric objects. So it turns out that to deform yourself means to choose a tangent direction on the sphere. There we found another strong link with moduli! Why on earth should we care about fat points?

Considering families over a fatter point, e. These are very different from the first order one, e. Good references are online notes by Ravi Vakil, and Sernesi's book Deformations of algebraic schemes. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 5 years, 4 months ago. Active 5 years, 4 months ago.

Viewed 3k times. 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.

May be I am missing some points for understanding. Babai Babai 4, 2 2 gold badges 20 20 silver badges 50 50 bronze badges. Everything is done in a special case and shown to follow from basic algebra. Active Oldest Votes. In general you have this: Definition. I'll tell you later what nice group describes these objects! Dori Bejleri 4, 1 1 gold badge 14 14 silver badges 20 20 bronze badges. Brenin Brenin Ravi's introductory lecture was probably good, although it has been a long time since I've watched it.

It's not in the link I gave above. Can you give any link for that "draft"? Thank you for your elaborate answer. Still many things are vague to me. May be as I read more I will understand it better. I guess in the process of understanding I will come up with more questions. I am not accepting the answer yet as someone might come up with a more illuminating answer. You are welcome to ask more questions as soon as you read more about it!

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## Deformation Theory - Graduate Texts in Mathematics

It seems that you're in Germany. We have a dedicated site for Germany. The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck. The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley. This introduction to deformation theory is based on his notes for a course he taught in

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## Deformation Theory

Buy now. Delivery included to Germany. Due to the Covid pandemic, our despatch and delivery times are taking a little longer than normal. Read more here. Robin Hartshorne Hardback 10 Dec English. Includes delivery to Germany. Check for new and used marketplace copies.

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The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck. The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley. Skip to main content Skip to table of contents. Advertisement Hide. This service is more advanced with JavaScript available.