REGULARITY FUNCTIONS OF POWERS OF GRADED IDEALS

Time:

Venue/Location: A204, VIASM

Speaker: Prof. Ngô Việt Trung ( Viện Toán học)

Abstract: This talk discusses 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. These functions are asymptotically linear with slope d. If dim R/I = 0, we give a complete description 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.