In the case of Riemann surfacesone can explain that the complex structure on the Riemann edformation is isolated no moduli. Sign up or log in Sign up using Google. To motivative the definition of haetshorne pre-deformation functor, consider the projective hypersurface over a field.
I am just writing my comment as an answer. Seshadri and b Cohomology of certain moduli spaces of vector bundles Proc. This website uses cookies to improve your experience while you navigate through the website. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are as essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website.
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This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information. Skip to content. Search for:. This website uses cookies to improve your experience. Another application of deformation theory is with Galois deformations.
The general Kodaira—Spencer theory identifies as the key to the deformation theory the sheaf cohomology group. In the case of Riemann surfacesone can explain that the complex structure on the Riemann sphere is isolated no moduli. Infinitesimals have long been in use by mathematicians for non-rigorous arguments in calculus. This was put on a firm basis by foundational work of Kunihiko Kodaira and Donald C. One of the major applications of deformation theory is in arithmetic.
I am just writing my comment as an answer. For example, in the geometry 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. For genus 1, an elliptic curve has a one-parameter family of complex structures, as shown in elliptic function theory. I am not accepting the answer yet as someone might come up with a more illuminating answer. 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.
It can be used to answer the following question: Then, the space on the right hand corner is one example of an infinitesimal deformation: This is true for moduli of curves. Why on earth should we care about fat points? This website uses cookies to improve your experience while you navigate through the website.
Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are as essential for the working of basic functionalities of the website. Replacing C by one of the components has the effect of decreasing either the genus or the degree of C.
But I have no clue how. If we have a Galois representation. Since we want to consider all possible expansions, we will let our predeformation functor be defined on objects as. And by the way there is another error on the same page, line -1, there is a -2 that should be a I think the harttshorne you mentioned is the following one: I do not have the book in front of me, but it sounds to me hartsshorne the formulation above is false.
In other words, deformations are regulated by holomorphic quadratic differentials on a Riemann surface, again something known classically.
Post as a guest Name. So it turns out that to deform yourself means to choose a tangent direction on the sphere. I have tried reading few lecture notes, for example: Mathematics Stack Exchange works best theogy JavaScript enabled. The intuition is that we want to study the infinitesimal structure haftshorne some moduli space around a point where lying above that point is the space of interest.
I am just writing my comment as an answer. Some characteristic phenomena are: I would appreciate if someone writes an answer either stating 1 Why to study deformation theory? The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties.
Since we are considering the tangent space of a point of some moduli space, we can define the tangent hartsjorne of our pre -deformation functor as. This website uses cookies to improve your experience while you navigate through the website. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are as essential for the working of basic functionalities of the website.
We also use third-party cookies that help us analyze and understand how you use this website.
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