# On the Hilbert function of general fat points in $\mathbb{P}^1 \times \mathbb{P}^1$

3 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
Abstract : We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A.V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat points in standard projective spaces. In this way, we compute the entire bi-graded Hilbert function in the case of triple points.
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Cited literature [9 references]

https://hal.inria.fr/hal-01637942
Contributor : Oneto Alessandro <>
Submitted on : Saturday, November 18, 2017 - 5:55:21 PM
Last modification on : Thursday, November 26, 2020 - 3:50:03 PM
Long-term archiving on: : Monday, February 19, 2018 - 12:48:18 PM

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• HAL Id : hal-01637942, version 1

### Citation

Enrico Carlini, Maria Catalisano, Alessandro Oneto. On the Hilbert function of general fat points in $\mathbb{P}^1 \times \mathbb{P}^1$. 2017. ⟨hal-01637942⟩

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