Glicci ideals

A central problem in liaison theory is to decide whether every arithmetically Cohen-Macaulay subscheme of projective $n$-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can be indeed achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an $(n+1)$-dimensional projective space. For example, this result applies to all reduced arithmetically Cohen-Macaulay subschemes. We also show that every union of fat points in projective 3-space can be linked in the same space to a union of simple points in finitely many steps, and hence to a complete intersection in projective 4-space.Comment: 8 page

Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

The regularity of points in multi-projective spaces

Let I = p_1^{m_1} \cap ... \cap p_s^{m_s} be the defining ideal of a scheme of fat points in P^{n_1} x ... x P^{n_k} with support in generic position. When all the m_i's are 1, we explicitly calculate the Castelnuovo-Mumford regularity of I. In general, if at least one m_i >= 2, we give an upper bound for the regularity of I, which extends the result of Catalisano, Trung and Valla to the multi-projective case.Comment: 12 pages with minor revisions. To appear in JPA

Separators of fat points in P^n

In this paper we extend the definition of a separator of a point P in P^n to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z = m_1P_1 + ... + m_iP_i + .... + m_sP_s in P^n and Z' = m_1P_1 + ... + (m_i-1)P_i + .... + m_sP_s. We associate to P_i a tuple of positive integers of length v = deg Z - deg Z'. We call this tuple the degree of the minimal separators of P_i of multiplicity m_i, and we denote it by deg_Z(P_i) = (d_1,...,d_v). We show that if one knows deg_Z(P_i) and the Hilbert function of Z, one will also know the Hilbert function of Z'. We also show that the entries of deg_Z(P_i) are related to the shifts in the last syzygy module of I_Z. Both results generalize well known results about reduced sets of points and their separators.Comment: 22 pages; minor revisions throughout; to appear in Journal of Algebr