The asymptotic behaviour of symbolic generic initial systems of generic points (Q392427)

Studying the asymptotic invariants of graded families of ideals has been an interesting subject in commutative algebra. The generic initial ideals which are coordinate-independent share many algebraic and geometric information with the initial ideals. Moreover, the symbolic generic initial system is a graded system of ideals.NEWLINENEWLINEFor an arbitrary graded sequence of zero-dimensional ideals, the multiplicity of the sequence is equal to its volume, and the multiplicity of an ideal \(I\) can be interpreted in terms of the multiplicities of the initial monomial ideals of the powers of \(I\) (see [\textit{L. Ein} et al., Am. J. Math. 125, No. 2, 409--440 (2003; Zbl 1033.14030)]). Let \(P_I\) denote the Newton polytope of the monomial ideal \(I\). Then for a graded system of monomial ideals \(\{a_q\}_q\) the polytopes \(\{\frac{1}{q}P_{a_q}\}_q\) are nested, and the limiting shape \(P\) of \(\{a_q\}_q\) is defined as the limit of the polytopes in this set. In case that the ideals of \(a_q\) are all zero-dimensional, the closure \(Q_q\) of each set \(\mathbb{R}_{\geq 0}^n\backslash P_{a_q}\) in \(\mathbb{R}^n\) is compact and by a result of \textit{M. Mustaţǎ} [J. Algebra 256, No. 1, 229--249 (2002; Zbl 1076.13500)] \({\text{vol}}(\{a_q\}_q)=n!{\text{vol}}(\bigcap_{q\in \mathbb{N}^\ast}\frac{1}{q}Q_{a_q})\).NEWLINENEWLINELet \(I \subset K[x,y,z]\) be the corresponding ideal to the points \(p_1,\ldots,p_r\) in \(\mathbb{P}^2\). The author studies the set of reverse lexicographic generic initial ideals of symbolic powers of \(I\). The main result of this paper describes the limiting shape of the symbolic generic initial system of \(I\) under assuming the Uniform Segre-Harbourne-Gimigliano-Hirschowitz Conjecture (for \(r\geq 9\)). Properties of the fat point ideals \(I^{(m)}\) are closely related to divisors on the blow-up of \(p_1,\ldots, p_r\) studied in [\textit{M. Nagata}, Mem. Coll. Sci., Univ. Kyoto, Ser. A 33, 271--293 (1960; Zbl 0100.16801)].