2. Divisors and line bundles ........................ 99. 2.1. Divisors
.............................. 99. 2.2. Line bundles
............................ 100. 2.3. Sections of line bundles
....................... 101. 2.4. The Riemann-Roch Theorem
..................... 103. 2.5. Line bundles and embeddings in
projective space ............ 105. 2.6. Hyperelliptic curves
......................... 106. 3. Abelian varieties
............................ 108. 3.1. Complex tori and Abelian
varieties .................. 108. 3.2. Line bundles on Abelian varieties
................... 109. 3.3. Abelian surfaces
.......................... 111. 4. Jacobi varieties
............................. 114. 4.1. The algebraic Jacobian
....................... 114. 4.2. The analytic/transcendental Jacobian
................. 114. 4.3. Abel's Theorem and Jacobi inversion
................. 119. 4.4. Jacobi and Kummer surfaces
..................... 121. 5. Abelian surfaces of type (1,4)
....................... 123. 5.1. The generic case
.......................... 123. 5.2. The non-generic case
........................ 124. V. Algebraic completely integrable
Hamiltonian systems ........ 127. 1. Introduction
.............................. 127. 2. A.c.i. systems
............................. 129. 3. Painlev analysis for a.c.i,
systems .................... 135. 4. The linearization of
two-dkmensional a.e.i, systems ............. 138. 5. Lax equations
............................. 140. VI. The Mumford systems
..................... 143. 1. Introduction
.............................. 143. 2. Genesis
................................ 145. 2.1. The algebra of
pseudo-differential operators .............. 145. 2.2. The matrix
associated to two commuting operators ........... 146. 2.3. The inverse
construction ....................... 150. 2.4. The KP vector fields
........................ 152. ix 3. Multi-Hamiltonian structure and
symmetries ................ 155. 3.1. The loop algebra 9(q
........................ 155. 3.2. Reducing the R-brackets and the
vector field ............. 157. 4. The odd and the even Mumford systems
.................. 161. 4.1. The (odd) Mumford system
..................... 161. 4.2. The even Mumford system
...................... 163.
