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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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