Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the 10-web by conics on the quartic del Pezzo surface

Luc Pirio

Abstract

We study and compare the webs $\boldsymbol{\mathcal W}_{{\rm dP}_d}$ defined by the conic fibrations on a given smooth del Pezzo surface ${\rm dP}_d$ of degree $d$ for $d=4$ and $d=5$. In a previous paper, we proved that for any positive $d\leq 6$, the web by conics $\boldsymbol{\mathcal W}_{{\rm dP}_d}$ carries a particular abelian relation ${\bf HLog}_d$, whose components all are weight $7-d$ antisymmetric hyperlogarithms. The web $\boldsymbol{\mathcal W}_{{\rm dP}_5}$ is a geometric model of the exceptional Bol's web and the relation ${\bf HLog}_5$ corresponds to the famous `Abel's identity' $(\boldsymbol{{\mathcal A}b})$ of the dilogarithm. Bol's web together with $(\boldsymbol{{\mathcal A}b})$ enjoy several remarkable properties of different kinds. We show that almost all of them admit natural generalizations to the pair $\big( \boldsymbol{\mathcal W}_{{\rm dP}_4}, {\bf HLog}_4\big)$.

On the 10-web by conics on the quartic del Pezzo surface

Abstract

We study and compare the webs defined by the conic fibrations on a given smooth del Pezzo surface of degree for and . In a previous paper, we proved that for any positive , the web by conics carries a particular abelian relation , whose components all are weight antisymmetric hyperlogarithms. The web is a geometric model of the exceptional Bol's web and the relation corresponds to the famous `Abel's identity' of the dilogarithm. Bol's web together with enjoy several remarkable properties of different kinds. We show that almost all of them admit natural generalizations to the pair .
Paper Structure (70 sections, 40 theorems, 259 equations, 2 figures, 5 tables)

This paper contains 70 sections, 40 theorems, 259 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Many remarkable properties of Bol's web
  3. Many remarkable properties of ${\mathcal{W}} \space{\mathcal{W}}_{{\bf dP}_4}$
  4. Structure of the paper
  5. Acknowledgements
  6. Hyperlogarithms, webs and del Pezzo surfaces in a nutshell
  7. Hyperlogarithms
  8. Webs
  9. Combinatorics of the space of abelian relations
  10. Hyperlogarithmic abelian relations
  11. Monodromies and residues
  12. Del Pezzo surfaces and their webs by conics
  13. Del Pezzo surfaces
  14. Lines and conics
  15. Some other properties of del Pezzo surfaces
  16. ...and 55 more sections

Key Result

Proposition 2.1

For $r=4,\ldots,8$, one has the following decompositions of $\mathbf C^{ \boldsymbol{\mathcal{L}}_r}$ in irreducible $W_r$-modules:See Hlog for some explanations about the notations used for the $W_r$-irreducibles.

Figures (2)

  • Figure 1: The marked Dynkin diagram of type $(D_5,\omega_4)$ on the left and the two possible types of rank four 1-marked Dynkin diagrams for the facets on the right: of type $(D_4,\omega_3)$ (up) and $(A_4,\omega_4)$ (bottom).
  • Figure 2: The marked Dynkin diagram associated to the spinor variety $\mathbb S_n^+={\rm Spin}_{2n}/P_{n-1}$.

Theorems & Definitions (60)

  • Proposition 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Remark 2.4
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 50 more