Postulation for 2-superfat points in the plane

TL;DR

This work analyzes the postulation of 0-dimensional schemes in the plane formed by unions of $2$-squares (the $2$-superfat points given by $(L_1^2,L_2^2)$) and proves they impose independent conditions on plane curves of degree $d$ up to the expected bound, i.e. $\dim I(X)_d=\max\{0,\binom{d+2}{2}-4s\}$. The authors develop Horace-method-based specializations, including line-aligned collisions and trace-residue control, to establish the main result for general unions of $2$-squares, and extend it to configurations including a general $3$-fat point, giving $\dim I(X_s)_d=\max\{\binom{d+2}{2}-4s-6,0\}$ except for an explicit exception. They then reinterpret these findings in an interpolation context, showing that the corresponding linear systems have geometrically interpretable double-point conditions (symmetric tangents or fixed double tangents) when positive. The methods provide a robust framework for analyzing postulation of 0-dimensional schemes built from fat points and their interpolation consequences in the plane.

Abstract

We study the postulation of 0-dimensional schemes given by unions of 2-superfat points in general position in the plane, i.e., the union of local schemes defined by the intersection of two distinct double lines. We prove that such schemes have good postulation, i.e., they have the expected Hilbert function. We also show the good postulation of such schemes when we add a general 3-fat point. Finally, we use these results to answer a peculiar kind of interpolation problem.

Figures (1)

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Theorems & Definitions (16)

• Definition 2.1
• Proposition 2.2
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8
• Remark 3.9