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oai:mountainscholar.org:10217/390372025-12-30T03:14:17Zcom_10217_100532com_10217_100000com_10217_100468com_10217_100303col_10217_100538col_10217_100469

Dumitrescu, Olivia Mirela, author Miranda, Rick, advisor Pries, Rachel, committee member Iyer, Hariharan K., committee member Peterson, Christopher Scott, 1963-, committee member 2007-01-03T05:44:44Z 2007-01-03T05:44:44Z 2010 Dumitrescu_colostate_0053A_10033.pdf ETDF2010100001MATH QA281 http://hdl.handle.net/10217/39037 https://doi.org/10.25675/3.018338 Department Head: Gerhard Dangelmayr. 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. born digital doctoral dissertations English eng Colorado State University. Libraries 2000-2019 Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. Techniques in interpolation problems Text Doctor of Philosophy (Ph.D.) Doctoral Mathematics Colorado State University This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). open.access