Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers

Charles Arnal

Abstract

In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension $n$ from families of such hypersurfaces in ambient dimensions $k=1,\ldots,n-1$. The asymptotic Betti numbers of real parts of the resulting family can then be described in terms of the asymptotic Betti numbers of the real parts of the families used as ingredients. The algorithm is based on Viro's Patchwork and inspired by I. Itenberg's and O. Viro's construction of asymptotically maximal families in arbitrary dimension. Using it, we prove that for any $n$ and $i=0,\ldots,n-1$, there is a family of asymptotically maximal real projective algebraic hypersurfaces $\{Y^n_d\}_d$ in $\mathbb{R} \mathbb{P} ^n$ (where $d$ denotes the degree of $Y^n_d$) such that the $i$-th Betti numbers $b_i(\mathbb{R} Y^n_d)$ are asymptotically strictly greater than the $(i,n-1-i)$-th Hodge numbers $h^{i,n-1-i}(\mathbb{C} Y^n_d)$. We also build families of real projective algebraic hypersurfaces whose real parts have asymptotic (in the degree $d$) Betti numbers that are asymptotically (in the ambient dimension $n$) very large.

Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers

Abstract

In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension from families of such hypersurfaces in ambient dimensions . The asymptotic Betti numbers of real parts of the resulting family can then be described in terms of the asymptotic Betti numbers of the real parts of the families used as ingredients. The algorithm is based on Viro's Patchwork and inspired by I. Itenberg's and O. Viro's construction of asymptotically maximal families in arbitrary dimension. Using it, we prove that for any and , there is a family of asymptotically maximal real projective algebraic hypersurfaces in (where denotes the degree of ) such that the -th Betti numbers are asymptotically strictly greater than the -th Hodge numbers . We also build families of real projective algebraic hypersurfaces whose real parts have asymptotic (in the degree ) Betti numbers that are asymptotically (in the ambient dimension ) very large.

Paper Structure

This paper contains 19 sections, 23 theorems, 157 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Viro's patchworking method
  3. The construction method
  4. Preliminaries
  5. A convex triangulation of Snd
  6. Choosing the coefficients of tildeQnd
  7. Defining Qnd using the Patchwork
  8. Computing asymptotic Betti numbers
  9. Preliminaries
  10. Finding cycles in a suspension
  11. Finding cycles in a join
  12. Counting cycles
  13. Asymptotically large Betti numbers in arbitrary dimension and index
  14. Asymptotic Hodge numbers and combinatorics
  15. Notations and known results
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $n\geq 2$. For $k=1,\ldots,n-1$, let $\{P^k_{d}\}_{d\in {\mathbb N}}$ be a family of completely nondegenerate real Laurent polynomials in $k$ variables, such that $P^k_d$ is of degree $d$ and that the Newton polytope $\Delta(P^k_d)$ of $P^k_d$ is $S^k_d$. Suppose additionally that for $k=1,\ldot for some $x_i^k \in {\mathbb R}_{\geq 0}$. Then there exists a family $\{Q^n_d\}_{d\in{\mathbb N}}$

Figures (13)

  • Figure 1: For $n-1 =3$, the sub-simplices of dimension $1,2$ and $3$ of $S^3_d$ that have to be maximal linearity domains of $\mu_d^{3}$ for it to satisfy the conditions of Lemma \ref{['LemmaConstructionsConvexSubdivision1']} are indicated in black and grey.
  • Figure 2: For $k=2$ and $d=6$, the graph of $\mu_{k-1}$ and the graph of $\mu$ on the face $\Gamma$ at the top, and the triangulation induced by $\mu$ on $\Gamma$ below.
  • Figure 3: In black, the subsets $R^{3}_{6,0,1}$, $R^{3}_{6,1,1}$, $R^{3}_{6,2,1}$ and $R^{3}_{6,3,1}$ of $S^3_6$ on the left and the subsets $R^{3}_{6,0,2}$, $R^{3}_{6,1,2}$ and $R^{3}_{6,2,2}$ on the right.
  • Figure 4: The joins (respectively, the cones) that have to appear in the triangulation $T$ for it to satisfy the conditions of Lemma \ref{['LemmaConstructionsConvexSubdivision2']} are shown in black and grey on the left (respectively, the right).
  • Figure 5: For $n=3$ and $d=6$: on the left, the subdivision $\tilde{T}$ restricted to $S^3_{6,0+}$, on the right, the subdivision $\tilde{T}$ restricted to $S^3_{6,1+}$.
  • ...and 8 more figures

Theorems & Definitions (60)

  • Theorem 1.1: Cooking Theorem
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.1: Main Patchwork Theorem
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 50 more