Ngo Viet Trung (Vietnam Academy of Science and Technology)Title: Regularity functions of graded ideals Abstract: This talk presents new results on the problem which sequences of non-negative integers arise as the functions reg I^(n-1)/I^n, reg R/I^n, reg I^n for an ideal I generated by forms of degree d in a standard graded algebra R (jointly obtained with Le Tuan Hoa and Nguyen Dang Hop). These functions are asymptotically linear with slope d. If dim R/I = 0, we give a complete characterization of the functions reg I^(n-1)/I^n, reg R/I^n and show that reg I^n can be any numerical function f(n) >= dn that weakly decreases first and then becomes linear. The latter result gives a negative answer to a question of Eisenbud and Ulrich. If dim R/I >= 1, we show that reg I^(n-1)/I^n, can be any numerical asymptotically linear function f(n) >= dn-1 and reg R/I^n can be any numerical asymptotically linear function f(n) >= dn-1 that is weakly increasing. We also prove that the function of the saturation degree of I^n is asymptotically linear for any graded ideal I. As a consequence, there exists a linear bound for this function which is asymptotically optimal and better than a recent result of Ein, Ha and Lazarsfeld for ideals cut out nonsingular projective schemes.
