Borel Fixed Initial Ideals of Prime Ideals in Dimension Two

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger. Let R = k[x1, x2, . . . , xr] be a polynomial ring over a field. We will say that an ideal I ⊆ R has the saturated chain property if given any non-minimal associated prime ideal Q of I there exists an associated prime ideal P ⊆ Q such that dim(R/P ) = dim(R/Q)+1. Therefore given Q an associated prime ideal of I, there exists a saturated chain of associated prime ideals P1 ⊂ P2 ⊂ · · ·Pn = Q such that P1 is minimal and dim(R/Pi) = dim(R/Pi+1) + 1 for 1 ≤ i ≤ n − 1. In 1999 Hosten and Thomas [HT] proved that the initial ideal of a toric ideal has this saturated chain property. This type of connectivity does not exist in general for initial ideals of prime ideals as the following counter-example, provided by Hosten and S. Popescu, illustrates. Counter Example 1. Take the toric ideal 〈xz−a, yz−b, tz−c〉 and substitute z−t for t, making the ideal 〈xz−a, yz−b, z−tz−c〉 which is still a prime ideal. Using the reverse lexicographic order and the variable order x > y > z > t > a > b > c the initial ideal of this prime ideal is J = 〈z, yz, xz, zb, za, ytb, xtb, xta〉. The primary decomposition of J is J = 〈x, y, z〉 ∩ 〈y, t, z〉 ∩ 〈t, z, a〉 ∩ 〈z, a, b〉 ∩ 〈x, y, z, a, b〉. Hence Ass(R/J) = {〈x, y, z〉, 〈y, t, z〉, 〈t, a, z〉, 〈z, a, b〉, 〈x, y, z, a, b〉}. Hosten and others have since constructed families of prime ideals such that the initial ideal of a prime ideal in the family does not have the saturated chain property. However, it was conjectured that lexicographic generic initial ideals of homogeneous prime ideals have the saturated chain property. We prove that if P is a homogeneous prime ideal of R = k[x1, x2, . . . , xr], dim(R/P ) = 2 and the initial ideal of P is Borel-fixed, then the initial ideal of P has the saturated chain property. There are many prime ideals with Borel-fixed initial ideal since for any prime ideal P the generic initial ideal of P is Borel-fixed. While we restrict the dimension, we do not make any assumptions on the monomial order used, nor do we require that the initial ideal be generic, only Borel-fixed which is weaker. We collect some key definitions and properties and then give the main theorem. A monomial order ≥ on a polynomial ring R = k[x1, x2, . . . , xr] over a field k is a total order on the monomials in R such that m ≥ 1 for each monomial m in R and if m1, m2, n are monomials in R with m1 ≥ m2 then nm1 ≥ nm2. A The research of the author was partially supported by the National Security Agency.