On bigraded regularities of Rees algebra

For any homogeneous ideal I in (K[x_1,ldots ,x_n]) of analytic spread (ell ), we show that for the Rees algebra R(I), ({text {reg}}_{(0,1)}^{mathrm{syz}}(R(I))={text {reg}}_{(0,1)}^{mathrm{T}}(R(I))). We compute a formula for the (0, 1)-regularity of R(I), which is a bigraded analog of Theorem 1.1 of Aramova and Herzog (Am. J. Math. 122(4) (2000) 689–719) and Theorem 2.2 of Römer (Ill. J. Math. 45(4) (2001) 1361–1376) to R(I). We show that if the defect sequence, (e_k:= {text {reg}}(I^k)-krho (I)), is weakly increasing for (k ge {text {reg}}^{mathrm{syz}}_{(0,1)}(R(I))), then ({text {reg}}(I^j)=jrho (I)+e) for (j ge {text {reg}}^{mathrm{syz}}_{(0,1)}(R(I))+ell ), where (ell ={text {min}}{mu (J)~|~ Jsubseteq I text{ a } text{ graded } text{ minimal } text{ reduction } text{ of } I}). This is an improvement of Corollary 5.9(i) of [16].