  We study syzygies of the Segre embedding of P(V_1) x ... x P(V_n), and prove
two finiteness results. First, for fixed p but varying n and V_i, there is a
finite list of "master p-syzygies" from which all other p-syzygies can be
derived by simple substitutions. Second, we define a power series f_p with
coefficients in something like the Schur algebra, which contains essentially
all the information of p-syzygies of Segre embeddings (for all n and V_i), and
show that it is a rational function. The list of master p-syzygies and the
numerator and denominator of f_p can be computed algorithmically (in theory).
The central observation of this paper is that by considering all Segre
embeddings at once (i.e., letting n and the V_i vary) certain structure on the
space of p-syzygies emerges. We formalize this structure in the concept of a
Delta-module. Many of our results on syzygies are specializations of general
results on Delta-modules that we establish. Our theory also applies to certain
other families of varieties, such as tangent and secant varieties of Segre
embeddings.
