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We will generalize this technique and using toric geometry methods, we prove the interpolation problems for triple points. Using non-toric degenerations we prove the emptiness of a linear system with ten multiple points for different ratios, a result that approximates from below Nagata's bound by rational numbers. In the introduction we also state other results obtained and we mention different directions for further research.","abstract_html":"This dissertation studies degeneration techniques in interpolation problems, that can be phrased as computing the dimension of the space of plane curves of degree d having general multiple points. The general interpolation problem goes back to the origin of algebraic geometry and is still far from being solved. We approach it using algebraic geometry techniques, by systematically exploiting degenerations of the projective plane. Degenerating the plane into a union of planes we prove the planar case of the interpolation problem for double points, and we present results obtained for higher multiplicities. We will generalize this technique and using toric geometry methods, we prove the interpolation problems for triple points. Using non-toric degenerations we prove the emptiness of a linear system with ten multiple points for different ratios, a result that approximates from below Nagata's bound by rational numbers. In the introduction we also state other results obtained and we mention different directions for further research.","abstract_has_math":false,"creators":["Dumitrescu, Olivia Mirela, author","Miranda, Rick, advisor","Pries, Rachel, committee member","Iyer, Hariharan K., committee member","Peterson, Christopher Scott, 1963-, committee member"],"institution":"Colorado State University. 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We approach it using algebraic geometry techniques, by systematically exploiting degenerations of the projective plane. Degenerating the plane into a union of planes we prove the planar case of the interpolation problem for double points, and we present results obtained for higher multiplicities. We will generalize this technique and using toric geometry methods, we prove the interpolation problems for triple points. Using non-toric degenerations we prove the emptiness of a linear system with ten multiple points for different ratios, a result that approximates from below Nagata's bound by rational numbers. 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