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''Jacobian, Jacobian variety, of an algebraic curve $S$''

The principally polarized
[[Abelian variety|Abelian variety]] (cf. also
[[Polarized algebraic variety|Polarized algebraic variety]]) $(J(S),\Theta)$
formed from this curve. Sometimes a Jacobi variety is simply
considered to be a commutative
[[Algebraic group|algebraic group]]. If $S$ is a smooth projective
curve of genus $g$ over the field $\C$, or, in classical terminology, a
compact
[[Riemann surface|Riemann surface]] of genus $g$, then the integration
of holomorphic $1$-forms over the $1$-cycles on $S$ defines the
imbedding 
$$H_1(S,\Z)\to H^0(S,\Omega_S)^*,$$
the image of which is a lattice of maximal rank (here
$\Omega_S$ denotes the bundle of holomorphic $1$-forms on $S$). The Jacobi
variety of the curve $S$ is the quotient variety 
$$J(S) = H^0(S,\Omega_S)^*/H_1(S,\Z).$$
For the
polarization on it one can take the cohomology class $\Theta$ from 
$$H^1(J(S),\Z)\land H^1(J(S),\Z) = H^2(J(S),\Z)\subset H^2(J(S),\C)$$
that corresponds to the intersection form on $H_1(S,\Z) \cong H_1(J(S),\Z)$. This polarization is
principal, that is, $\Theta^g=g'$. For a more explicit definition of a Jacobi
variety it is usual to take a basis $\delta_1,\dots,\delta_{2g}$ in $H_1(S,\Z)$ and a basis of forms
$\omega_1,\dots,\omega_g$ in $H^0(S,\Omega_S)$. These define a $(g\times 2g)$-matrix $\Omega$ - the matrix of periods of
the Riemann surface: 
$$\Omega = ||\int_{\delta_j}\omega_j||.$$
Then $J(S)=\C^g/\Lambda$, where $\Lambda$ is the lattice with
basis consisting of the columns of $\Omega$. The bases $\delta_j$ and $\omega_i$ can be
chosen so that $\Omega = ||E_g Z||$; here the matrix $Z=X+iY$ is symmetric and $Y>0$ (see
[[Abelian differential|Abelian differential]]). The polarization class
is represented by the form $\omega$ that, when written in standard
coordinates $(z_1,\dots,z_g)$ in $\C^g$, is 
$$\omega = \frac{i}{2} \sum_{1\le j,k\le g} (Y^{-1})_{jk}dz_j\land d{\bar z}_k.$$
Often, instead of the cohomology
class $\Theta$ the effective
[[Divisor|divisor]] dual to it is considered; it is denoted by the
same letter and is defined uniquely up to a
translation. Geometrically, the divisor $\Theta$ can be described in the
following way. Consider the Abelian mapping $\mu:S\to J(S)$ defined by 
$$\mu(s) = \big(\int_{s_0}^s \omega_1,\dots,\int_{s_0}^s \omega_g\big)+\Lambda,$$
where
$s_0\in S$ is fixed. Let $S^{(d)}$ be the $d$-th symmetric power of $S$, that is,
the quotient variety of the variety $S^d$ with respect to the symmetric
group (the points of $S^{(d)}$ correspond to effective divisors of degree
$d$ on $S$). The formula $\mu(s_1,\dots,s_d) = \mu(s_1)+\cdots + \mu(s_d)$ defines an extension of the Abelian
mapping to $\mu:S^{(d)}\to J(S)$. Then $\Theta=W_{g-1} = J(S)$.

The equivalence relation in $S^{(g)}$ defined by $\mu$ coincides with the
rational equivalence of divisors (Abel's theorem). In addition, $\mu(S^{(g)}) = J(S)$
(Jacobi's inversion theorem). C.G.J. Jacobi
studied the inversion problem in the case $g=2$ (see also
[[Jacobi inversion problem|Jacobi inversion problem]]). The
above-mentioned theorems determine an isomorphism $J(S)\cong {\rm Pic}^g(S)$, where ${\rm Pic}^g(S)$ is
the component of the
[[Picard group|Picard group]] ${\rm Pic}(S)$ corresponding to divisors of degree
$g$. Multiplication by the divisor class $-gs_0$ leads to a canonical
isomorphism $J(S)\cong {\rm Pic}^0(S)$ of Abelian varieties.

In the case of a complete smooth curve over an arbitrary field, the
Jacobi variety $J(S)$ is defined as the
[[Picard variety|Picard variety]] ${\rm Pic}(S)$. The Abelian mapping $\mu$
associates with a point $s\in S$ the class of the divisor $s-s_0$, and the
polarization is defined by the divisor $W_{g-1}=\mu(S^{(g-1)})$.

The significance of Jacobi varieties in the theory of algebraic curves
is clear from the Torelli theorem (cf.
[[Torelli theorems|Torelli theorems]]): A non-singular complete curve
is uniquely defined by its Jacobian (with due regard for polarization)
(see
[[#References|[Griff[1]]]). The passage from a curve to its Jacobian enables
one to linearize a number of non-linear problems in the theory of
curves. For example, the problem of describing special divisors on $S$
(that is, effective divisors $D$ for which $H^0(S,O(K-D))>0$) is essentially
translated to the language of singularities of special subvarieties
$W_d=\mu(S^{(d)})$ of $J(S)$. This translation is based on the Riemann–Kempf theorem
about singularities (see ,
[[#References|[Griff[1]]]). One of the corollaries of this theorem is that
the codimension of the variety of singular points of the divisor of
the polarization, $\Theta=W_{g-1}$, does not exceed 4. This property of Jacobi
varieties is characteristic if one considers only principally
polarized Abelian varieties belonging to a neighbourhood of the
Jacobian of a general curve. More precisely, if the variety of
singular points of the divisor of the polarization of a principally
polarized Abelian variety $A$ has codimension $\le 4$, and if $A$ does not
belong to several distinguished components of the moduli variety, then
$A\cong J(S)$ for a smooth curve $S$ (see
[[#References|[AM]]]).

Another approach to distinguishing Jacobians among Abelian varieties
is to define equations in $\theta$-functions and their derivatives at
special points. The problem of finding these equations is called
Schottky's problem.

In the case of a singular curve $S$ the Jacobi variety $J(S)$ is regarded
as the subgroup of ${\rm Pic}(S)$ 
defined by divisors of degree 0 with respect to
each irreducible component of $S$ (it coincides with the connected
component of the identity in ${\rm Pic}(S)$). If the curve $S$ is defined by a
module ${\frak m}$ on a smooth model $N$, then $J(S)$ is usually called the
generalized Jacobian of the curve $N$ (relative to ${\frak m}$), and is
denoted by $J_{\frak m}$ (see
[[#References|[Se]]]).

====References====
(For sorting, please click on [sort] below.) 

{| class="sortable" 
|- 
! scope="col" | [sort]
! scope="col" class="unsortable" | 
|-
|valign="top"|[Jac1]||valign="top"| C.G.J. Jacobi, "Considerationes generales de transcendentibus abelianis" ''J. Reine Angew. Math.'', '''9''' (1832) pp. 349–403 {{ZBL|009.0357cj}}  {{ZBL|14.0314.01}}
|-
|valign="top"|[Jac2]||valign="top"| C.G.J. Jacobi, "De functionibus duarum variabilium quadrupliciter periodicis, quibus theoria transcendentium abelianarum innititur" ''J. Reine Angew. Math.'', '''13''' (1835) pp. 55–78 {{ZBL|013.0473cj}} {{ZBL|26.0506.01}} {{ZBL|14.0314.01}}
|-
|valign="top"|[AM]||valign="top"| A. Andreotti, A. Mayer, "On period relations for abelian integrals on algebraic curves" ''Ann. Scu. Norm. Sup. Pisa'', '''21''' (1967) pp. 189–238 {{MR|0220740}} {{ZBL|0222.14024}}
|-
|valign="top"|[Griff2]||valign="top"| P.A. Griffiths, "An introduction to the theory of special divisors on algebraic curves", Amer. Math. Soc.  (1980) {{MR|0572270}} {{ZBL|0446.14010}}
|-
|valign="top"|[Mum]||valign="top"| D. Mumford, "Curves and their Jacobians", Univ. Michigan Press (1978) {{MR|0419430 }}
|-
|valign="top"|[Griff1]||valign="top"| P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry", Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}}
|-
|valign="top"|[Se]||valign="top"| J-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959) {{MR|0103191 }}
|-
|}

====Comments====
The Schottky problem has been solved, cf.
[[Schottky problem|Schottky problem]].

Here, a module on a smooth curve $N$ is simply an effective divisor,
i.e., a finite set $S$ of points of $N$ with a positive integer
$\nu_P$ assigned to each point $P\in S$. 
Given a module $\frak m $ and a rational function
$g$ on $N$, one writes $g\equiv 1 \mod {\frak m}$ 
if $1-g$ has a zero of order $\ge\nu_P$ in $P$
for all $P\in S$. Consider divisors $D$ whose support does not intersect
$S$. For these divisors one defines an equivalence relation: $D_1\sim_{\frak m}D_2$ if
there is a rational function $g$ such that $(g) = D_1 - D_2$ and $g\equiv 1 \mod {\frak m}$. This is
the equivalence relation that serves to define the generalized
Jacobian $J_{\frak m}$, cf.
[[#References|[Se]]], Chapt. V for details. In general, the generalized
Jacobian is not complete; it is an extension of $J(N)$ by a connected
linear algebraic group. Every Abelian extension of the function field
of $N$ can be obtained by an
[[Isogeny|isogeny]] of a generalized Jacobian. This is a main reason
for studying them,
[[#References|[Se]]].

In the case of an arbitrary field the construction of the Jacobi
variety $J(S)$ of a complete smooth curve $S$ was achieved by A. Weil,
first as an abstract algebraic variety (see
[[#References|[a1]]] and
[[#References|[a2]]]), and later as a projective variety by W.L. Chow
(see
[[#References|[a3]]]).

For the theory of the singularities of the $\theta$-divisor and for the
Torelli theorem see also
[[#References|[a4]]].

====References====
{| class="sortable"
|- 
! scope="col" | [sort]
! scope="col" class="unsortable" | 
|-
|valign="top"|[Weil]||valign="top"| A. Weil, "Courbes algébriques et variétés abéliennes. Variétés abéliennes et courbes algébriques", Hermann (1971)
|-
|valign="top"|[Lang]||valign="top"| S. Lang, "Abelian varieties", Springer (1981)
|-
|valign="top"|[Chow]||valign="top"| W.L. Chow, "The Jacobian variety of an algebraic curve" ''Amer. J. Math.'', '''76''' (1954) pp. 453–476
|-
|valign="top"|[ACHG]||valign="top"| E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves", '''1''', Springer (1985)
|-
|}