Date of Graduation

5-2025

Document Type

Thesis

Degree Name

Bachelor of Science in Mathematics

Degree Level

Undergraduate

Department

Mathematical Sciences

Advisor/Mentor

Mantero, Paolo

Committee Member

Wheeler, Jill

Second Committee Member

Bergdall, John

Third Committee Member

D'Eugenio, Daniela

Abstract

Given some set of r general points in the projective plane, we want to better understand: what is the smallest degree of any polynomial passing through the points m times? How many linearly independent equations of this degree pass through the points m times? The investigation of these questions, particularly for the case of m=3 and r< 16, motivates the development of several results. We translate Terracini's inductive argument, a tool for evaluating the expectedness of certain sets of double points, into a version which can be used for triple points, and prove that the argument holds. We compute the minimal graded free resolutions for the ideals corresponding to up to 15 points, for m up to 6, and we conjecture a connection between the expectedness of these ideals and what their resolutions look like. Further, we prove that this conjecture holds when m=1, and we either fully or partially prove that these ideals are expected for certain sets of r general triple points, with r up to 13. These results are obtained using a variety of tools from commutative algebra, in addition to computations using Macaulay2.

Keywords

Commutative Algebra; Algebraic Geometry

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