Interpolation on surfaces in ℙ³

Huizenga, Jack

doi:10.1090/S0002-9947-2012-05582-6

Interpolation on surfaces in $\mathbb {P}^3$
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Trans. Amer. Math. Soc. 365 (2013), 623-644 Request permission

Abstract:

Suppose $S$ is a surface in $\mathbb {P}^3$, and $p_1,\ldots ,p_r$ are general points on $S$. What is the dimension of the space of sections of $\mathcal {O}_S(e)$ having singularities of multiplicity $m_i$ at $p_i$ for all $i$? We formulate two natural conjectures which would answer this question, and we show they are equivalent. We then prove these conjectures in case all multiplicities are at most $4$.

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