This is a post about me reading the Introduction of Chapter 9 in the book FGA Explained. In particular, it is about the history of Picard schemes, an ubiquitous object in algebraic geometry.

Roughly speaking, this is the underlying scheme of the Picard group of , where for simplicity we can assume to be integral scheme projective over an algebraically closed field . This object was first introduced by Grothendieck (as you would expect) in 1962, but its history goes long before that.

Elliptic Integral

The Picard scheme has roots in 1600s, where Calculus was developed around the same time.

Bernoulli Family

In 1694, James Bernoulli analyzed the way rods bend, and was led to the vanishing set of with , which just look like the following


In polar coordinates, he found the arc length to be given by

$$s=\int_0^r\frac{a^2dr}{\sqrt{a^4-r^4}}$$

He concluded cannot be expressed in terms of elementary functions, and this is one instance of what’s now called elliptic integrals, because similar integrals already came up in attempts to rectify elliptical orbits.

In 1698, Jame’s brother, John Bernoulli, showed that given two arcs from the origin on the cubical parabola , their lengths differ by the length of a certain third such arc. He posed the problem of finding more cases of this phenomenon.

Fagnano, Euler

Between 1714 and 1720, Fagnano found similar relations for cords and arcs of ellipses, hyperbolas and lemniscates (the figure eights above). Based on this, Euler in 1757 found the addition formula

$$\int_0^{x_1}\frac{dx}{\sqrt{1-x^4}}\pm \int_0^{x_2}\frac{dx}{\sqrt{1-x^4}}=\int_0^{x_3}\frac{dx}{\sqrt{1-x^4}}$$

where the variables must satisfy certain symmetric relation.

In 1759, Euler generalized this formula to some other elliptic integrals, and he expressed regret he could handle only square roots and fourth power, but no higher roots or powers.

Abel

In 1826, Abel discovered an addition theorem for whats now called Abelian integrals. They are algebraic integrals of the following form

$$\psi x:=\int_{x_0}^xR(x,y)dx$$

where is an independent complex variable, a rational function, and is an integral algebraic function, i.e. implicit multivalued function defined by an irreducible equation where

$$f(x,y):=y^n+f_1(x)y^{n-1}+...+f_n(x)$$

where the are polynomials in .

Let be the genus of the curve , and be rational numbers. Then Abel’s addition theorem says that

$$\sum_{i=1}^p h_i\psi x_i=v+\sum_{i=1}^g \psi x_i'$$

where is an elementary function of the independent variables $x1,…,x_px_1’,…,x_g’gg-1x_1’,…,x{g-1}’x1,…,x_p\sum{i=1}^p hi\psi x_i\sum{i=1}^{g-1}\psi x_i’$.

Despite the fact his paper on this major result has Cauchy and Legendre as referees, the Royal Academy of Science did not publish it until 1841, which is long before Abel’s death on 1829.

Meanwhile, Abel summarized his general addition theorem informally in Crelle’s Journal, where he treated in detail of a major special case, namely where is a square-free polynomial of degree . In particular, he found if is odd and if is even. Thus, if then , and hence Euler’s formula cannot possible extend.

Jacobi, Legendre

With Jacobi’s help, Legendre came to appreciate this particular case. He recovered the case for , and for , since the integrals share many formal properties as , he named those integrals “ultra-elliptiques”.

Later Jacobi translated “ultra-elliptiques” by “hyperelliptischen”, and the prefix “hyper” has stuck eversince. In Legendre’s cover letter, he praised Abel’s addition theorem “a monument more lasting than bronze”.

A few month later, Jacobi proved Abel’s theorem in the hyperelliptic case, and posed the famous problem, now known as “Jacobi Inversion Problem”. It asks to give a characterization of functions whose inverse are Abelian integrals, and what does Abel’s addition theorem show about them.

In 1827, Abel had originated the idea of inverting elliptic integrals, obtaining what’s known as “elliptic functions”. He and Jacobi studied them extensively, and in the process Jacobi introduced “theta functions” as an aid in the study. They were generalized by Riemann in 1857, and used to solve the Jacobi inversion problem in any genus.

Abel, Again

In a 1829 paper, Abel addressed two intermediate questions. First, when is the sum constant? Second, what is the number of that can vary independently? For general case, Abel did provide a computation, but answer was not very definitive. He showed constancy holds when is of the form

$$\psi x=\int_{x_0}^x \frac{h(x,y)}{\partial f/\partial y}dx$$

where . Also, must satisfy certain linear conditions, namely it must vanish suitably everywhere does on the curve (this should remind you of divisors).

Further, Abel found there exists such that

$$n-\alpha=g-i$$

where are as in the beginning of this subsection.

Algebraic Geometry

Riemann

Abel’s idea have been clarified and completed over the course of time, and doubtlessly Riemann made the greatest contribution in this 1857 paper on Abelian functions. He developed a way of extending complex analysis to a multivalued function of a single variable by viewing as a single-valued function on an abstract multisheeted covering of the -plane, i.e. the Riemann surface of . At the time, Riemann’s theory was strange and suspect, as at that time we don’t even have theory of manifolds.

Riemann defined the genus topologically, as half of the first Betti number of the surface. However, the term “genus” is not coined by Riemann, but Clebsch.

Anyhow, Riemann defined an integral to be of the “first kind” if it is finite everywhere. He proved these integrals form a vector space of dimension . Then, by fixing a basis and a homology basis of paths. Now, inside the vector space , he formed the lattice generated by the corresponding -vectors of periods, and proved the quotient is a -dimensional complex torus

$$J:=\mathbb{C}^g/\Lambda$$

Later was termed the “Jacobian” to honour Jacobi’s work on inversion.

Associated with the Jacobian was the Abel-Jacobi map, which is defined by

$$\Psi_n:\mathrm{Sym}^n(C)\to J,\quad (x_1,...,x_n)\mapsto \left(\sum_{i=1}^n\psi_1x_i,...,\sum_{i=1}^n\psi_gx_i\right)$$

Riemann studied the fibers of the Abel map, and proved that if two divisors and are linearly equivalent, then .

The converse claim also holds, and was stated by Clebsch in 1864 and proved some time later by Weierstrass, i.e. then the two divisors are linearly equivalent. Together, we see the fiber is the complete linear system determined by , where the dimension of this space was predicted by Abel (it is in the previous section).

Furthermore, Riemann rediscovered Abel’s formula , and in 1864, Roch identified as the number of independent adjoints vanishing on . In the same year, Brill and Noether gave the first algebro-geometric treatment of the formula, and named the statement as “Riemann-Roch Theorem”.

Finally, Riemann treated the inversion problem proposed by Jacobi. He proved that the Abel map is biholomorphic on certain saturated Zariski open subset , where the inverse map can be expressed using the coordinate functions on , so in terms of functions on . Since , these functions can be lifted to an open subset of , and then continued to meromorphic functions on with periods.

Those functions are called “Abelian functions”, and it turns out the set of all Abelian functions whose group of periods contains a given lattice in of rank is a field of transcendence degree over . That is, is the field of rational functions on a -dimensional projective algebraic variety , which is parameterized on a Zariski open set by of them.

Those varities were all called “Abelian varieties” at first, and it was shown by Lefschetz there is a distinguished whose underlying set can be identified with in a natural way, and he restricted the term “Abelian variety” to it.

Castelnuovo, Clebsch, M. Nother

In the case of a curve , its Jacobian is an abelian variety. That is, and are defined by integrals, but can be described by polynomials. In particular, addition on corresponds to addition of divisors.

In 1868, Clebsch generalized Abel’s formula to surface , with ordinary singularities and no point at infinity on the -axis. Clebsch showed every double integral of the first kind is of the form

$$\int\int \frac{h(x,y,z)}{\partial f/\partial z}dxdy$$

where and vanishes when does. The number of independent integrals became known as the geometric genus and denoted by . In 1871, Cayley found a formula for the expected number of independent , and called it the arithmetic genus, and denoted by .

In 1875, Max Noether explained the discrepancy between and : the vanishing condition on need not be independent. In any case, .

In 1897, Castelnuovo fixed a linear system of curves on the surface , and studied the “characteristic” linear system cut out on a general member by the other members

To be continued…