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Matrixwise (approach to unexpected hypersurfaces) Reloaded

Marcin Dumnicki, Grzegorz Malara, Halszka Tutaj-Gasińska

TL;DR

The authors introduce a purely geometric-combinatorial framework to certify the existence of degree-$d$ hypersurfaces in $\mathbb{P}^n$ with a general point of multiplicity $m$ that vanish on a given point set $Z$, by studying the determinant $F$ of an interpolation matrix $M$ built from $Z$ and derivatives up to order $m-1$. The central result provides a lower bound on the multiplicity of any point $B$ in the zero set $\{F=0\}$ in terms of weak combinatorics $h_{j,B}$ derived from the linear systems $\mathcal{L}(d;jB+Z)$. Through a sequence of concrete examples in $\mathbb{P}^2$ and $\mathbb{P}^3$, including square and rectangular interpolation matrices and configurations such as $A(4k+1,1)$, Fermat configurations, and the $D4$ setup, they demonstrate many new instances where $F\equiv 0$, yielding unexpected curves or hypersurfaces; the approach connects to known dualities and circumvents exhaustive computer-based computations by exploiting combinatorial incidences and Bézout-type bounds. The results broaden the toolkit for proving the existence of unexpected hypersurfaces and highlight a rich interplay between weak combinatorics and algebraic geometry.

Abstract

In the paper we provide a new method of proving the existence of a hypersurface of degree $d$ in $\mathbb{P}^n$, with a general point of multiplicity $m$ and vanishing at a given set of points $Z$, by looking at weak combinatorics of a set $Z$. This method has a direct application in the theory of unexpected hypersurfaces, where many of the examples are based only on computer experiments.

Matrixwise (approach to unexpected hypersurfaces) Reloaded

TL;DR

The authors introduce a purely geometric-combinatorial framework to certify the existence of degree- hypersurfaces in with a general point of multiplicity that vanish on a given point set , by studying the determinant of an interpolation matrix built from and derivatives up to order . The central result provides a lower bound on the multiplicity of any point in the zero set in terms of weak combinatorics derived from the linear systems . Through a sequence of concrete examples in and , including square and rectangular interpolation matrices and configurations such as , Fermat configurations, and the setup, they demonstrate many new instances where , yielding unexpected curves or hypersurfaces; the approach connects to known dualities and circumvents exhaustive computer-based computations by exploiting combinatorial incidences and Bézout-type bounds. The results broaden the toolkit for proving the existence of unexpected hypersurfaces and highlight a rich interplay between weak combinatorics and algebraic geometry.

Abstract

In the paper we provide a new method of proving the existence of a hypersurface of degree in , with a general point of multiplicity and vanishing at a given set of points , by looking at weak combinatorics of a set . This method has a direct application in the theory of unexpected hypersurfaces, where many of the examples are based only on computer experiments.

Paper Structure

This paper contains 6 sections, 5 theorems, 52 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Examples with a square interpolation matrix
  4. Examples with a rectangular interpolation matrix
  5. Example of unexpected curves of type $(d+k, d)$, $k>1$
  6. In $\mathbb{P}^3$

Key Result

Theorem 2.3

Let $d$ and $m$ be positive integers with $d\geq m$. Let $P_1,\dots, P_s$ be pairwise distinct points in $\mathbb{P}^N$, where $s$ is such that Let $B$ be a fixed point in $\mathbb{P}^N$, and $F(a_0,\ldots,a_N)=\det M$ a polynomial of degree $\binom{m+N-1}{N}(d-m+1)$. Define and $h_B:=\sum_{j=1}^d h_{j,B}.$ Then the multiplicity of $B$ in the set $F=0$ is at least $h_B$.

Figures (2)

  • Figure 1: The set $\{F=0\}$ for 7 and for 9 general points
  • Figure 2: In the adjacent figure, the points from the set $F_6$ are represented as black filled dots, while the points from $F_3$ are shown as red open circles. The points belonging to the set $T$ are depicted as blue squares. Additionally, for a representative point $P$, the lines $l_1, l_2, l_3$ passing through $P$ are also illustrated in the figure.

Theorems & Definitions (29)

  • Definition 1.1: HMNT
  • Remark 2.1
  • Example 2.2
  • Theorem 2.3
  • proof
  • Remark 2.5
  • Example 3.1
  • Remark 3.2
  • Example 3.3
  • Remark 3.4
  • ...and 19 more