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Generators and resolutions of ideals defining certain surfaces in projective space
Abstract
Let X be the blow up of the points of transverse intersection of plane curves P and Q. Let $F\sb{d,m}$ be the divisor class on X corresponding to forms on ${\bf P}\sp2$ of degree d vanishing at each point of $P\cap Q$ to order at least m. We consider the question of finding a minimal set of generators for the ideal $I\sb{X}$ defining the image of X mapped to projective space via the sections of such a divisor class. Geramita, Gimigliano and Harbourne (GGH) answer this question for the case d = t + 1, m = 1, when P and Q have the same degree t. We find a minimal set of generators for $I\sb{X}$ defining X embedded in ${\bf P}\sp{N}$ by the sections of any very ample $F\sb{d,m}$, regardless of the degrees of P and Q (and in particular, not assuming P and Q have the same degree). In case P and Q have the same degree, one can arrange for the embedding $X \to {\bf P}\sp{N}$ to factor as X $\subset {\bf P}\sp1 \times {\bf P}\sp2 \to \ {\bf P}\sp{N}$. We show that $I\sb{X}$ is minimally generated in at most two degrees: by quadrics which cut out ${\bf P}\sp1 \times \ {\bf P}\sp2$ in ${\bf P}\sp{N}$, and by a remaining set of generators which then cut out X. We also work out the general case, in which the degrees of P and Q are not necessarily the same. The result here is more complicated, with generators occurring in various degrees. Moreover, we consider the more general question of finding all the graded Betti numbers in a minimal free resolution of $I\sb{X}$. For those cases that (GGH) determines generators, we also determine the graded Betti numbers in a minimal free resolution. All of this work holds in arbitrary characteristic. We also describe a question about the geometry of certain bigraded rings and make a conjecture about the problem of finding minimal free resolutions in characteristic zero using Schur functors.
Subject Area
Mathematics
Recommended Citation
Holay, Sandeep H, "Generators and resolutions of ideals defining certain surfaces in projective space" (1994). ETD collection for University of Nebraska-Lincoln. AAI9507813.
https://digitalcommons.unl.edu/dissertations/AAI9507813