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# Ideal Containments Under Flat Extensions and Interpolation on Linear Systems in P2

#### 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*[**P*** ^{ n}*],

*I*

*⊆*

^{(mn)}*I*for all

^{m}*m*∈

**N.**Over the projective plane, we obtain

*I*

^{(4)}⊆

*I*

^{2}. Huneke asked whether it was the case that

*I*

^{(3)}⊆

*I*

^{2}. 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)}⊆

*I*

^{2}. 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)}⊆

*I*

^{2}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

*q*be a set of not necessarily general points and let

_{1}, . . ., q_{ r}*p*be a set of general points in

_{1}, . . ., p_{s}**P**

^{2},

*r*+

*s*≤ 8. Let

*X*be a blow up of the points with

*e*

_{ 1}, . . .,

*e*and

_{r}*E*, . . . ,

_{1}*E*the corresponding exceptional curves. Write

_{s}*e*=

*a*

_{1}

*e*

_{ 1}+ · · · +

*a*and

_{r}e^{r}*E*=

*b*

_{1}

*E*

_{ 1}+ · · · +

*b*For the two linear systems [

_{s}E_{s}.*dL – e – E*] and [

*dL – e*] with [

*dL – e – E*] ⊆ [

*dL – e]*, we give a condition sufficient to guarantee that

**[special characters omitted]**and another condition necessary for

**[special characters omitted]**When

*r*= 7,

*s*= 1,

*d*= 3,

*a*= 1, 1 ≤

_{j}*j*≤ 7 and

*b*= 2, we connect the discussion to quasi-elliptic fibrations and show that when

*q*

_{ 1}+ · · · +

*q*

_{7}is reduced, then

*h*

^{0}(

*X,*3

*L*–

*e*

_{1}– · · · –

*e*

_{7}– 2

*E*) > max{0,

*h*

^{0}(

*X,*3

*L*–

*e*

_{1}– · · ·–

*e*

_{7}) – 3} if and only if

*q*

_{ 1}+ · · · +

*q*

_{7}is the union of the seven points of the Fano plane. Allowing infinitely near points, we obtain nonreduced subschemes

*q*

_{1}+ · · · +

*q*

_{7}, consisting of essentially distinct points, that form part of the base loci of quasi-elliptic fibrations such that

*h*

^{0}(

*X,*3

*L*–

*e*

_{1}– · · · –

*e*

_{7}– 2

*E*) > max{0,

*h*

^{ 0}(

*X,*3

*L*–

*e*

_{ 1}– · · · –

*e*

_{ 7}) – 3}.^

#### Subject Area

Mathematics

#### Recommended Citation

Akesseh, Solomon, "Ideal Containments Under Flat Extensions and Interpolation on Linear Systems in P2" (2017). *ETD collection for University of Nebraska - Lincoln*. AAI10608382.

https://digitalcommons.unl.edu/dissertations/AAI10608382