## Posts tagged ‘algebraic geometry’

### Elliptic curves for high school students

I had to give a talk to high school students about some mathematical notion: I decided to tell them something about elliptic curves, but not the usual speech about cryptography, finite fields and the group law on a cubic curve…

Instead, I talked about the perhaps less known appearances of elliptic functions as solutions of classical ODEs (even if I don’t really know much about these myself). The simplest mechanical system whose motion is governed by an elliptic curve is the pendulum: the reason for this is that the ODE $\ddot{x} + \sin x = 0$ which classically describes the time evolution of the angle of the pendulum is best rewritten in terms of the altitude of the pendulum: the law of energy conservation is then written as
$p^2 = q(q-q_0)(q-2l) = P(q)$
where 0 and 2l are the extremal values of the altitude $q$, $q_0$ is the highest altitude which can be reached with a given energy (even if $q_0 > 2l$, which corresponds to the pendulum make full rotations around its axis), and $p$ is the vertical momentum of the pendulum.

In this setting, there are classical Hamilton relations $dq = p dt$, $dp = P'(q) dt$, so the differential form $dt = dq/p$ turns out to be the canonical non-vanishing abelian differential on the elliptic curve. This explains why the period of the pendulum is an elliptic integral, which can be calculed by an arithmetic-geometric mean, and why the position of the pendulum at $t = t_1 + t_2$ can be deduced from its position at times $t_1$ and $t_2$ by the classical secant-tangent law.

The notes for the talk (in French) are available here.

### Schemes in algebraic geometry 3 : glued schemes and sheaves

André Weil was among the first ones to point out the importance of having a local description of varieties, especially projective spaces, which can always locally be described as an affine space with completion by a hyperplane at infinity, and projective varieties, which similarly look like varieties in affine space. The use of sheaves in local description of spaces was magnified by Cartan and Serre, in the context of complex analytic spaces, and generalised to the algebraic setting by Serre in Faisceaux algébriques cohérents.

The projective space is the simplest example of an algebro-geometric object which cannot be described by the prime spectrum or the functor of points of a ring. For example, there is no obvious ring whose ideals describe varieties in projective space, which come from homogeneous equations. We would like to give a correct definition of gluing affine lines (with coordinates $z$ and $1/z$) to define the projective line $\mathbb P^1$ as the gluing of $\mathbb A^1$ with $\mathbb A^1 \to \mathbb P^1$ given by $z \mapsto 1/z$. For functors of points, the latest article by Alain Connes and Caterina Consani, gives a definition. For prime spectra, one has to be aware that gluing only topological spaces do not give meaningful information on algebraic properties. This is illustrated by the case of differentiable manifolds, which are not the same as topological manifolds: gluing differentiable manifolds has to induce a correspondance between differentiable functions (this is equivalent to the requirement that gluing maps between charts be differentiable).
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### Schemes in algebraic geometry 2 : prime spectra and generic points

I just explained how the affine plane could be described by the ring $\mathbb Z[x,y]$. A point M of the affine plane whose coordinate ring is R is a morphism $\mathbb Z[x,y] \to R$ defined by the assignment $P \mapsto P(a,b) \in R$, where $(a,b)$ are the coordinates of M. In the case of points corresponding to morphisms $\mathbb Z[x,y] \to \mathbb Z$, there is a natural way of recovering the point from the ring morphism by looking at his equations, which are elements of the kernel of the morphism. If M satisfies the equations $x=a$ and $y=b$, then M has the form $(a,b)$. This motivates the abstract definition of point of the affine plane as a morphism $\mathbb Z[x,y] \to R$ to some ring.

Conversely, the set of equations of M defines a canonically associated point $p_M$, which is the morphism $\mathbb Z[x,y] \to \mathbb Z[x,y]/I$, where $I$ is the ideal generated by the equations. But this morphism has no reason to totally recover M if it wasn’t a point with integral coordinates. For example, the point $(2,3)$ is a special point, satisfying a lot of equations, which characterize it. But $(\log 2, \pi)$ do not satisfy any polynomial equations with integral coefficients, so the set of its equations is empty, and cannot be used to recover it. Moreover, the point $(e, \log 3)$ does not satisfy equations either: their algebraic properties are exactly the same. These points are called generic.

The prime spectrum of a ring is a convenient way of describing equivalence classes of points of a given ring.
Definition. The prime spectrum of $A = \mathbb Z[x,y]$ is the set of points $p_I: A \to A/I$ for prime ideals $I$. If $M: A \to R$ is any point of the affine plane with coordinates in an integral domain $R$, then Mis canonically associated to some $p_M := p_I$, where $I$ is the kernel of the map $A \to R$.
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### Schemes in algebraic geometry I : the affine plane

I think most people blogging around algebraic geometry eventually write about schemes, (as in Rigorous trivialities or algebraic stacks (in the Secret Blogging Seminar), which are traditionnally seen as the main reason (not) to study algebraic geometry today. My turn now. I recommend Igor Dolgacev’s lectures which is one of my favorite ways of speaking of schemes.

Interesting mathematics come up when algebraic varieties (things defined by several polynomial equations) are no longer defined as mere sets (sets of tuples of numbers satisfying the equation) but mope complex mathematical objects. Differential geometry, for example, gives the structure of a complex manifold to algebraic varieties in $\mathbb C^n$, which is still an efficient way of proving theorems. However, during the 20th century, a lot of mathematicians tried to develop a new structure which would avoid the use of analysis to concentrate on the algebraic aspects (I don’t know exaclty who, but expect Hilbert, Zariski, Chevalley, Grothendieck to have played a role). Grothendieck approach using category theory and functors of points is now widely used and is a very impressive way of tying together intuition, commutative algebra, and geometry.

There are many ways of reverse engineering Grothendieck’s definition of a scheme (see EGA1 if you want to know how this is related to Chevalley’s definition of a scheme). The first thing to say is probably what properties and notions are needed for schemes:
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### Monads in mathematics 4 : the bar and cobar constructions

The use of monads and comonads in homological algebra is as old as the theory: Godement’s standard construction refers to the use of monads in cohomology theory, and is said to be the first study of a general method for constructing acyclic resolutions. Later the theory was explored in greater depth by Eilenberg, Moore, Barr, Beck. The term bar resolution is now most commonly used to describe the process. Monads derived from operads were also studied by Lawvere, Mitchell, Bénabou, under the name of algebraic theories, with aim towards topoi and logic.

Bar resolutions (see J. Baez website and the LNM Seminar on Triples and Categorical Homology Theory) are a way to canonically (i.e. functorially) describe an arbitrary algebra over a monad using only free algebras. It is a particular case of definition with generators and relations which is often the only way to describe infinite mathematical objects with finite expressions (computer algebra systems usually deal with finitely generated objects, and use generators and relations to describe their elements), but bar resolutions somehow are the universal way of doing this.

Most cohomology theories fit in the framework of bar constructions, though in various apparently unrelated ways. However, a visible common denominator of most constructions is the use of simplicial methods. This makes them of some use in homotopical algebra: bar resolutions are used to define canonical cofibrant resolutions of objects, which explains their uses in definition of derived functors.
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### Experimental algebraic geometry I : the grassmannian

I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.

Macaulay provides a command-line interface using the readline library (like many other command-line programs) : here is what input/output looks like

i1 : V = Grassmannian(1,3)
o1 = ideal(p   p    - p   p    + p   p   )
1,2 0,3    0,2 1,3    0,1 2,3
o1 : Ideal of ZZ[p   , p   , p   , p   , p   , p   ]
0,1   0,2   1,2   0,3   1,3   2,3


When I type a command at i1, I get an output o1 with a value $\mathrm{ideal}(p_{1,2} p_{0,3} - p_{0,2} p_{1,3} + p_{0,1} p_{2,3})$ and a type: this output is an ideal of the ring $\mathbb{Z}[p_{0,1}, p_{0,2}, p_{1,2}, p_{0,3}, p_{1,3}, p_{2,3}]$. Many features of the Grassmannian as an algebraic variety are available: first define

i1 : V = Grassmannian(1,3,CoefficientRing => QQ);
o1 : Ideal of QQ[p   , p   , p   , p   , p   , p   ]
0,1   0,2   1,2   0,3   1,3   2,3
i2 : X = Proj(ring V / V)
o2 = X
o2 : ProjectiveVariety


the projective variety X over $\mathbb Q$ defined by the homogeneous ideal V : here ring V denotes the ambient ring of V. We see that $V$ is a non-singular quadric in 5-dimensional projective space, and check several well-known facts (more…)

### Lines in space and the Grassmann variety

This is an English version of my previous post. Schubert calculus is a collection of techniques and formulae used for computations of enumerative, numerical properties of common objects in linear algebra (there are very nice books by William Fulton covering the subject). The example of the set of lines in three-dimensional space is commonly chosen to illustrate the kind of results which are obtained by Schubert calculus: the reason is probably that it is the simplest non trivial situation.

A line in affine space would usually by defined by its direction (which is a line in a vector space, and depends on two parameters: imagine parameterising it by a vector on a sphere), and its position, depending on two more parameters, for a given direction (think of the possible translations of a given line). Thus the set of lines might be parameterised by four parameters, and more precisely one can show that it has dimension four. Even more interestingly, two parallel planes define for almost every line a unique pair of intersection points (one on each plane) which enables us to recover the line. This means lines are (almost all) parameterised by four coordinates, by means of rational functions (well, we still have to explain what is parameterised: for example, we could say that the slope, direction of the line and its position are rational functions of the coordinates on the two planes). The resulting parameterisation is somewhat bijective, but a few lines are missing (for example, lines parallel to the plane): however, we say the set of lines is a rational variety: which means the parameterisation is essentially bijective, and the missing elements can be obtained by taking limits.

### The Plücker embedding

To avoid tiring case disjunctions, it is very nice to extend the theory to the set of lines in projective space. They can be defined by their Plücker coordinates : given a line in space, we choose two points, given by projective (a.k.a. barycentric) coordinates ${M = [a_1:a_2:a_3:a_4]}$ and ${N = [b_1:b_2:b_3:b_4]}$. Then we define the Plücker coordinates of the line $MN$ by the six numbers $x_{ij} = (a_i b_j - a_j b_i)$. The fundamental properties of these coordinates is their invariance under changing the choice of points. Any other pair of points would be given by barycentres of M and N (with weights $(\lambda,\mu)$ and $(\nu,\pi)$), giving new coordinates
$(\lambda a_i + \mu b_i)(\nu a_j + \pi b_j) - (\lambda a_j + \mu b_j)(\nu a_i + \pi b_i)$
$= (\lambda \pi - \mu \nu)(a_i b_j - a_j b_i)$
which are actually proportional to the old coordinates. These coordinates, as projective coordinates, are thus uniquely defined (and even allow to recover the line). The set of lines now appears as a subvariety of a projective space called the Grassmann variety of lines in 3-space.