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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The infinitesimal conditions are deformatjon the result of applying the approach of differential calculus to solving a problem with constraints. 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. Some characteristic phenomena are: In some form these considerations have a history of centuries in mathematics, but also in physics and engineering.
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. Perturbation theory also looks at deformations, in general of operators.
Seminar on deformations and moduli spaces in algebraic geometry and applications
The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties. This was put on a firm basis by foundational work of Kunihiko Kodaira and Donald C. Spencerafter deformation techniques had received a great deal of more tentative application in the Italian school dsformation algebraic geometry. One expects, intuitively, that deformation theory of the first order should equate the Zariski tangent space with a moduli space.
The phenomena turn out to be rather subtle, though, in the general case. In the case of Riemann surfacesone can explain that the complex structure on the Riemann sphere is isolated no moduli. For genus 1, an elliptic curve has a one-parameter family of complex structures, as shown in elliptic function theory. The general Kodaira—Spencer theory identifies as the key to the deformation theory the sheaf cohomology group. 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.
In the case of genus 0 the H 1 vanishes, also. For genus defformation the dimension is the Hodge number h 1,0 which is therefore 1. These obviously depend on theor parameters, a and b, whereas the isomorphism classes of such curves have only one parameter.
Hence there must be an equation relating those a and b which describe isomorphic elliptic curves. In other words, deformations are regulated by holomorphic quadratic differentials on a Riemann surface, again something known classically.
These examples are the beginning of a theory applying to holomorphic families of complex manifolds, of any dimension. Another method for formalizing deformation theory is using functors on the category of local Artin algebras over a field. A pre-deformation functor is defined as a functor. The intuition is that we want to study the infinitesimal structure of some moduli space around a point where lying above that hartshornd is the space of interest. It is typically edformation case that it is easier to described the functor for a moduli problem instead of finding an actual space.
This is true for moduli of curves. Infinitesimals have long been in use by mathematicians for non-rigorous arguments in calculus. A simple application of this is that we can find the derivatives of monomials using infinitesimals:.
We could also interpret this equation as the first two terms of the Taylor expansion of the monomial. Infinitesimals can be made rigorous using nilpotent elements in local artin algebras. For our monomial, suppose we want to write out the second order expansion, then.
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. To motivative the definition of a pre-deformation functor, consider the projective hypersurface over a field. If we want to consider an infinitesimal deformation hartshofne this space, then we could write down a Cartesian square.
Then, the space on the right hand corner is one example of an infinitesimal deformation: Since we want to consider all possible expansions, we will let our predeformation functor reformation defined on objects as.
Since we are considering the tangent space of a point of some moduli space, we can define the tangent space of our pre -deformation functor as. Deformation theory was famously applied in birational geometry by Shigefumi Mori to study the existence of rational curves on varieties. The method of the proof later became known as Mori’s bend-and-break. The rough idea is to start with some curve C through a chosen point and keep deforming it until it breaks into several components.
Replacing C by one of the components has the effect of decreasing either the genus or the degree of C.
algebraic geometry – Studying Deformation Theory of Schemes – Mathematics Stack Exchange
So after several repetitions of the procedure, eventually we’ll obtain a curve of genus 0, i. The existence and the properties of deformations of C require arguments from deformation theory and a reduction to positive characteristic.
One of detormation major applications of deformation theory is in arithmetic. It can be used to answer the following question: Another application of deformation theory is with Galois deformations. It allows deforjation to answer the question: If we have a Galois representation. 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.