Ideal Containments under Flat Extensions and Interpolation on Linear Systems in P²

Authors

Solomon Akesseh, University of Nebraska-LincolnFollow

ORCID IDs

Solomon Akesseh

First Advisor

Brian Harbourne

Date of this Version

8-2017

Document Type

Article

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Brian Harbourne. Lincoln, Nebraska: August, 2017

Copyright (c) 2017 Solomon Akesseh

Abstract

Fat points and their ideals have stimulated a lot of research but this dissertation concerns itself with aspects of only two of them, broadly categorized here as, the ideal containments and polynomial interpolation problems.

Ein-Lazarsfeld-Smith and Hochster-Huneke cumulatively showed that for all ideals I in k[Pⁿ], I^(mn) ⊆ I^m for all m ∈ N. Over the projective plane, we obtain I⁽⁴⁾< ⊆ I². Huneke asked whether it was the case that I⁽³⁾ ⊆ I². Dumnicki, Szemberg and Tutaj-Gasinska show that if I is the saturated homogeneous radical ideal of the 12 points of the Hesse configuration, then I⁽³⁾ ̸⊆ I². Since then, additional examples have been found, but all of them, are the intersection loci of lines. Here we extend all the examples of I⁽³⁾ ̸⊆ I2 to points that are not directly the intersection loci of lines but are the intersection loci of curves.

In the case of the interpolation problem, this dissertation makes the following contribution. Let k be an algebraically closed field of arbitrary characteristic. Let q₁,...,q_r be a set of not necessarily general points and let p₁,...,p_s be a set of general points in P², r+s ≤ 8. Let X be a blow up of the points with e₁,...,e_r and E₁, . . . , E_s the corresponding exceptional curves. Write e = a₁e₁ + · · · + a_re_r and E = b₁E₁ +···+b_sE_s. For the two linear systems [dL−e−E] and [dL−e] with [dL − e − E] ⊆ [dL − e], we give a condition sufficient to guarantee that

h0(X, dL − e − E) > max{0, h0(X, dL − e) − ∑((bi+1) choose 2)} and another condition necessary for h⁰(X, dL − e − E) > max{0, h⁰(X, dL − e) − ∑((bi+1) choose 2)}. When r =7,s=1,d=3,a_j =1,1≤j≤7 and b=2, we connect the discussion to quasi-elliptic fibrations and show that when q₁ + · · · + q₇ is reduced, then h⁰(X,3L−e₁ −···−e₇ −2E) > max{0,h⁰(X,3L−e₁ −···−e₇)−3} if and only if q₁ + · · · + q₇ is the union of the seven points of the Fano plane. Allowing infinitely near points, we obtain nonreduced subschemes q₁ + · · · + q₇, consisting of essentially distinct points, that form part of the base loci of quasi-elliptic fibrations such that h⁰(X,3L−e₁ −···−e₇ −2E) > max{0,h⁰(X,3L−e₁ −···−e₇)−3}.

Adviser: Brian Harbourne