The Limiting Polytope of the Generic Initial System of a Complete Intersection

The asymptotic behaviour of symbolic generic initial systems of generic points

“…The results presented here and in [May12a] may lead the reader to believe that the limiting polytope of any symbolic generic initial system is defined by a single hyperplane. The following example shows that this does not hold even for ideals of points in P 2 .…”

“…We will see that each of the ideals gin(I (m) ) is generated in the variables x and y, so that P gin(I (m) ) , and thus P , can be thought of as a subset of R 2 . One reason for studying the limiting shape of a system of monomial ideals is that it completely determines the asymptotic multiplier ideals of the system (see [How01] and [May12a]).…”

“…When the point arrangement has an ideal I that is a complete intersection of type (α, β) with α ≤ β, a special case of the main result of [May12a] shows that the limiting shape of the symbolic generic initial system has a boundary defined by the line through the points (α, 0) and (0, β). The main result of this paper is the following theorem describing the limiting shape of the symbolic generic initial system of an ideal of r distinct points of P 2 in general position, assuming that the SHGH Conjecture 3.1 holds for the case where r ≥ 9.…”

“…Precisely what information is carried by the limiting shape of the symbolic generic initial system of other point arrangements is still uncertain. While one can prove that the x-intercept of the boundary of P is equal to r (I) (see Section 2), that the y-intercept reflects the asymptotic behaviour of the regularity of the ideals I (m) (see [May12a]), and that the volume under P is equal to r 2 (Proposition 2.14), there is likely additional geometric information encoded within P . Two important questions concern the form of P : is P always a polytope, and what does it mean for the boundary of P to be defined by a certain number of line segments?…”

The Symbolic Generic Initial System of Points on an Irreducible Conic

“…The asymptotic behaviour of a collection of monomial ideals a • such that a i • a j ⊆ a i+j (a graded system of monomial ideals) can be described by its limiting shape P . If P a i denotes the Newton polytope of a i , then the limiting shape P is defined to be the limit lim m→∞ 1 m P am ([May12d]). In addition to giving a simple geometric interpretation of the limiting behaviour, P completely determines the asymptotic multiplier ideals of a • (see [How01]).…”