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\centerline{\bf Giuseppe Valla}
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\centerline{\bf Castelnuovo Regularity and finiteness of Hilbert functions}
\bigskip{\bf Abstract}
\noindent The Castelnuovo-Mumford regularity is a kind of universal bound
for relevant invariants of graded algebras, such as the maximum degree of
the syzygies and the maximum non-vanishing degree of the local cohomology
modules. In this talk I will discuss a method to bound the regularity of
certain classes of standard graded algebras by means of the dimension
and any cohomological degree. This will be achieved through a purely
ring-theoretic version of a classical theorem of Mumford, concerning the
behaviour of the geometric regularity under generic hyperplane sections. We
will apply these ideas to prove some finiteness theorems for the number of
Hilbert Functions of certain classes of standard graded algebras. We will
give purely algebraic proofs of difficult results by Kleiman,
Srinivas-Trivedi, and more recently by Rossi-Valla-Vasconcelos and
Rossi-Trung-Valla.
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