On the Betti numbers and Rees algebras of ideals with linear powers

An ideal $$I \subset \mathbb {k}[x_1, \ldots , x_n]$$ I ⊂ k [ x 1 , … , x n ] is said to have linear powers if $$I^k$$ I k has a linear minimal free resolution, for all integers $$k>0$$ k > 0 . In this paper, we study the Betti numbers of $$I^k$$ I k , for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for $$d=2, 3$$ d = 2 , 3 or $$n-1$$ n - 1 , and the product of all ideals generated by s variables, for $$s=n-1$$ s = n - 1 or $$n-2$$ n - 2 . We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

Fibers of rational maps and Rees algebras of their base ideals

We consider a ratinonal map $\phi$ from m-dimensional projective space to n-dimensional projective space that is a parameterization of m-dimensional variety. Our main goal is to study the (m-1)-dimensional fibers of $\phi$ in relation with the m-th local cohomology modules of Rees algebra of its base ideal.

On Ratliff-Rush closure of modules

In this paper, we introduce the notion of Ratliff-Rush closure of modules and explore whether the condition of the Ratliff-Rush closure coincides with the integral closure. The main result characterizes the condition in terms of the normality of the projective scheme of the Rees algebra. In conclusion, we shall give a criterion for the Buchsbaum Rees algebras.

Fibers of rational maps and Rees algebras of their base ideals

We consider a ratinonal map $\phi$ from m-dimensional projective space to n-dimensional projective space that is a parameterization of m-dimensional variety. Our main goal is to study the (m-1)-dimensional fibers of $\phi$ in relation with the m-th local cohomology modules of Rees algebra of its base ideal.