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The Maximal Rank Conjecture for Sections...
Larson, Eric...

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The Maximal Rank Conjecture for Sections of Curves by Larson, Eric ( Author )
Lemonjar Open Access
Australian National University
27-07-2023
Let be a general curve of genus g embedded via a general linear series of degree d in P^r. The well-known Maximal Rank Conjecture asserts that the restriction maps H^0(O_{P^r}(m)) \to H^0(O_C(m) are of maximal rank; if known, this conjecture would determine the Hilbert function of C. In this paper, we prove an analogous statement for the hyperplane sections of unions general curves. More specifically, if H is a general hyperplane, we show that H^0(O_H(m)) \to H^0(O_{(C_1 \cup C_2 \cup \cdots \cup C_n) \cap H}(m)) is of maximal rank, except for some counterexamples when m = 2. As explained in arXiv:1809.05980, this result plays a key role in the author's proof of the Maximal Rank Conjecture.
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MYR 0.01
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http://arxiv.org/abs/1208.2730
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