Browse issues
no. 4
The 10-web by conics on the quartic del Pezzo surface
Luc Pirio1
1 LMV (Laboratoire de Mathématiques de Versailles), Univ. Versailles Saint-Quentin - Paris-Saclay & CNRS (UMR 8100), 78000 Versailles, France.
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 34 (2025) no. 4, pp. 1007-1146

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

Nous étudions et comparons les tissus $\mathcal{W}_{\mathrm{dP}_d}$ définis par les fibrations coniques sur une surface de del Pezzo lisse $\mathrm{dP}_d$ de degré $d$, pour $d=4$ et $d=5$. Dans un article précédent, nous avons prouvé que pour tout entier positif $d\le 6$, le tissu en coniques $\mathcal{W}_{\mathrm{dP}_d}$ sur une surface de del Pezzo $\mathrm{dP}_d$ de degré $d$ porte une relation abélienne particulière $\mathbf{HLog}_{{d}}$, dont toutes les composantes sont des hyperlogarithmes antisymétriques de poids $7-d$. Le tissu $\mathcal{W}_{\mathrm{dP}_5}$ est un modèle géométrique du tissu de Bol et la relation $\mathbf{HLog}_{{5}}$ correspond à la fameuse « identité d’Abel » $(\mathcal{A}b)$ du dilogarithme. Le tissu de Bol ainsi que la relation $(\mathcal{A}b)$ jouissent de plusieurs propriétés remarquables de différentes natures. Nous montrons que presque toutes ces propriétés admettent des généralisations naturelles à la paire $(\mathcal{W}_{\mathrm{dP}_4}, \mathbf{HLog}_{{4}})$.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1827
Classification: 53A60, 39B32, 14J26, 33E20, 14J45, 14L30
Keywords: Del Pezzo quartic surface, pencil of conics, web, abelian relation, hyperlogarithm
Mots-clés : Surface de del Pezzo quartique, pinceau de coniques, tissu, relation abélienne, hyperlogarithme

Luc Pirio 1

1 LMV (Laboratoire de Mathématiques de Versailles), Univ. Versailles Saint-Quentin - Paris-Saclay & CNRS (UMR 8100), 78000 Versailles, France.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2025_6_34_4_1007_0,
     author = {Luc Pirio},
     title = {The 10-web by conics on the  quartic del {Pezzo} surface},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1007--1146},
     year = {2025},
     publisher = {Universit\'e de Toulouse, Toulouse},
     volume = {Ser. 6, 34},
     number = {4},
     doi = {10.5802/afst.1827},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1827/}
}
TY  - JOUR
AU  - Luc Pirio
TI  - The 10-web by conics on the  quartic del Pezzo surface
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2025
SP  - 1007
EP  - 1146
VL  - 34
IS  - 4
PB  - Université de Toulouse, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1827/
DO  - 10.5802/afst.1827
LA  - en
ID  - AFST_2025_6_34_4_1007_0
ER  - 
%0 Journal Article
%A Luc Pirio
%T The 10-web by conics on the  quartic del Pezzo surface
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2025
%P 1007-1146
%V 34
%N 4
%I Université de Toulouse, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1827/
%R 10.5802/afst.1827
%G en
%F AFST_2025_6_34_4_1007_0
Luc Pirio. The 10-web by conics on the  quartic del Pezzo surface. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 34 (2025) no. 4, pp. 1007-1146. doi: 10.5802/afst.1827

[1] Hülya Argüz The quantum mirror to the quartic del Pezzo surface, Lie theory and its applications in physics (Springer Proceedings in Mathematics & Statistics), Volume 396, Springer, 2022, pp. 423-429 | DOI | MR | Zbl

[2] Michael F. Atiyah Convexity and commuting Hamiltonians, Bull. Lond. Math. Soc., Volume 14 (1982) no. 1, pp. 1-15 | DOI | MR | Zbl

[3] Dan Avritzer; Rick Miranda Stability of pencils of quadrics in P 4 , Bol. Soc. Mat. Mex., III. Ser., Volume 5 (1999) no. 2, pp. 281-300 | MR | Zbl

[4] Victor V. Batyrev; Oleg N. Popov The Cox ring of a del Pezzo surface, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002) (Progress in Mathematics), Volume 226, Birkhäuser, 2004, pp. 85-103 | DOI | MR | Zbl

[5] Vladimir G. Berkovich Integration of one-forms on p-adic analytic spaces, Annals of Mathematics Studies, 162, Princeton University Press, 2007, vi+156 pages | DOI | MR | Zbl

[6] Wilhelm Blaschke; Gerrit Bol Geometrie der Gewebe. Topologische Fragen der Differentialgeometrie, Grundlehren der Mathematischen Wissenschaften, 49, Springer, 1938 | MR | Zbl

[7] Gerrit Bol Topologische Fragen der Differentialgeometrie. LXIV. Über ein bemerkenswertes Fünfgewebe in der Ebene, Abh. Math. Semin. Univ. Hamb., Volume 11 (1936), pp. 387-393 | DOI | Zbl | MR

[8] Nicolas Bourbaki Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV, V & VI, Actualités Scientifiques et Industrielles, 1337, Hermann, 1968, 288 pages | DOI | MR

[9] Michel Brion; Claudio Procesi Action d’un tore dans une variété projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) (Progress in Mathematics), Volume 92, Birkhäuser, 1990, pp. 509-539 | MR | Zbl

[10] Francis Brown Single-valued hyperlogarithms and unipotent differential equations (2004) (unpublished text, https://www.ihes.fr/~brown/rhpaper5.pdf)

[11] Werner Burau Topologische Fragen der Differentialgeometrie. LXIII. Die Kegelschnittscharen einer Fläche 3. Ordnung, Abh. Math. Semin. Univ. Hamb., Volume 11 (1936), pp. 381-386 | DOI | Zbl | MR

[12] Ana-Maria Castravet; Luc Pirio Hyperlogarithmic functional equations on del Pezzo surfaces, Adv. Math., Volume 442 (2024), 109567, 25 pages | DOI | MR | Zbl

[13] Steven Charlton; Claude Duhr; Herbert Gangl Clean single-valued polylogarithms, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 17 (2021), 107, 34 pages | DOI | MR | Zbl

[14] Robert F. Coleman Dilogarithms, regulators and p-adic L-functions, Invent. Math., Volume 69 (1982) no. 2, pp. 171-208 | DOI | MR | Zbl

[15] Daniel F. Coray; Michael A. Tsfasman Arithmetic on singular Del Pezzo surfaces, Proc. Lond. Math. Soc. (3), Volume 57 (1988) no. 1, pp. 25-87 | DOI | MR | Zbl

[16] Daniel Corey Initial degenerations of spinor varieties, Eur. J. Math., Volume 10 (2024) no. 1, 4, 41 pages | DOI | MR | Zbl

[17] David B. Damiano Webs and characteristic forms of Grassmann manifolds, Am. J. Math., Volume 105 (1983) no. 6, pp. 1325-1345 | DOI | MR | Zbl

[18] Pierre Deligne Notes on spinors, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), American Mathematical Society, 1999, pp. 99-135 | MR | Zbl

[19] Ulrich Derenthal Universal torsors of del Pezzo surfaces and homogeneous spaces, Adv. Math., Volume 213 (2007) no. 2, pp. 849-864 | DOI | MR | Zbl

[20] Ulrich Derenthal Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc. Lond. Math. Soc. (3), Volume 108 (2014) no. 3, pp. 638-681 | DOI | MR | Zbl

[21] Igor V. Dolgachev Classical algebraic geometry. A modern view, Cambridge University Press, 2012, xii+639 pages | DOI | MR | Zbl

[22] Igor V. Dolgachev; Benson Farb; Eduard Looijenga Geometry of the Wiman-Edge pencil, I: algebro-geometric aspects, Eur. J. Math., Volume 4 (2018) no. 3, pp. 879-930 | DOI | MR | Zbl

[23] Benjamin Enriquez; Federico Zerbini Analogues of hyperlogarithm functions on affine complex curves (2024) | arXiv

[24] Vladimir V. Fock; Alexander B. Goncharov Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 6, pp. 865-930 | DOI | MR | Numdam | Zbl

[25] William Fulton; Joe Harris Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer, 1991, xvi+551 pages | DOI | MR

[26] Meinolf Geck; Götz Pfeiffer Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, 21, Clarendon Press, 2000, xvi+446 pages | MR | DOI | Zbl

[27] Israilʼ M. Gelʼfand; Robert D. MacPherson Geometry in Grassmannians and a generalization of the dilogarithm, Adv. Math., Volume 44 (1982) no. 3, pp. 279-312 | DOI | MR | Zbl

[28] Israilʼ M. Gelʼfand; Vera V. Serganova Combinatorial geometries and the strata of a torus on homogeneous compact manifolds, Usp. Mat. Nauk, Volume 42 (1987) no. 2(254), p. 107-134, 287 | MR | Zbl

[29] Mark Goresky; Robert D. MacPherson On the topology of algebraic torus actions, Algebraic groups Utrecht 1986 (Lecture Notes in Mathematics), Volume 1271, Springer, 1987, pp. 73-90 | DOI | MR | Zbl

[30] Richard M. Hain; Robert D. MacPherson Higher logarithms, Ill. J. Math., Volume 34 (1990) no. 2, pp. 392-475 | MR | Zbl

[31] William V. D. Hodge; Daniel Pedoe Methods of algebraic geometry. Vol. II. Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties, Cambridge University Press, 1952, x+394 pages | MR | Zbl

[32] Toshio Hosoh Automorphism groups of quartic del Pezzo surfaces, J. Algebra, Volume 185 (1996) no. 2, pp. 374-389 | DOI | MR | Zbl

[33] S. James Gates; S.-N. Hazel Mak; Marcus Spradlin; Anastasia Volovich Cluster Superalgebras and Stringy Integrals (2021) | arXiv

[34] Jae-Hyouk Lee Configurations of lines in del Pezzo surfaces with Gosset polytopes, Trans. Am. Math. Soc., Volume 366 (2014) no. 9, pp. 4939-4967 | DOI | MR | Zbl

[35] Travis Mandel Classification of rank 2 cluster varieties, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 15 (2019), 042, 32 pages | DOI | MR | Zbl

[36] Yuriĭ I. Manin Cubic forms. Algebra, geometry, arithmetic, North-Holland Mathematical Library, 4, North-Holland, 1986, x+326 pages | MR | Zbl

[37] Laurent Manivel Double spinor Calabi-Yau varieties, Épijournal de Géom. Algébr., EPIGA, Volume 3 (2019), 2, 14 pages | DOI | MR | Zbl

[38] Tomoki Nakanishi Dilogarithm identities in cluster scattering diagrams, Nagoya Math. J., Volume 253 (2024), pp. 1-22 | DOI | MR | Zbl

[39] Tomoki Nakanishi Pentagon relation in quantum cluster scattering diagrams (2024) | arXiv

[40] Soichi Okada Pfaffian formulas and Schur Q-function identities, Adv. Math., Volume 353 (2019), pp. 446-470 | DOI | MR | Zbl

[41] Jorge Vitório Pereira Resonance webs of hyperplane arrangements, Arrangements of hyperplanes—Sapporo 2009 (Advanced Studies in Pure Mathematics), Volume 62, Mathematical Society of Japan, 2012, pp. 261-291 | DOI | MR | Zbl

[42] Jorge Vitório Pereira; Luc Pirio An invitation to web geometry, IMPA Monographs, 2, Springer, 2015, xviii+213 pages | MR | DOI | Zbl

[43] Luc Pirio The abelian relations of minuscule Gelfand-MacPherson webs of type E r for r=4,...,8 (in preparation)

[44] Luc Pirio Équations fonctionnelles abéliennes et théorie des tissus, Ph. D. Thesis, Université Pierre et Marie Curie - Paris VI (2004) (https://theses.hal.science/tel-00335195)

[45] Luc Pirio On webs, polylogarithms and cluster algebras (2021) | arXiv | Zbl

[46] Luc Pirio Webs by conics on del Pezzo surfaces and hyperlogarithmic functional identities (2022) | arXiv | Zbl

[47] Luc Pirio On the (n+3)-webs by rational curves induced by the forgetful maps on the moduli spaces ℳ 0,n+3 , Diss. Math., Volume 584 (2023), pp. 1-133 | DOI | MR | Zbl

[48] Oleg N. Popov Del Pezzo surfaces and algebraic groups (2001) (Diplomarbeit, Universität Tübingen)

[49] Gilles Robert Relations fonctionnelles polylogarithmiques et tissus plans (2002) (prépublication 146, Université Bordeaux 1)

[50] Martin Schönert et al. GAP – Groups, Algorithms, and Programming – version 3 release 4 patchlevel 4 (1997) (1571 pp.)

[51] Corrado Segre Étude des différentes surfaces du quatrième ordre à conique double on cuspidale (générale ou décomposée) considérées comme des projections de l’intersection de deux variétés quadratiques de l’espace à quatre dimensions, Math. Ann., Volume 24 (1883), pp. 313-444 | DOI | MR | Zbl

[52] Corrado Segre Le linee principali di una superficie de S 5 e una proprietà caratteristica della superficie di Veronese. I, II, Rom. Acc. L. Rend. (5), Volume 30 (1921) no. 1, p. 200-203 & 227–231 | Zbl

[53] Vera V. Serganova; Alexei N. Skorobogatov Del Pezzo surfaces and representation theory, Algebra Number Theory, Volume 1 (2007) no. 4, pp. 393-419 | DOI | MR | Zbl

[54] Alexei N. Skorobogatov On a theorem of Enriques–Swinnerton-Dyer, Ann. Fac. Sci. Toulouse, Math. (6), Volume 2 (1993) no. 3, pp. 429-440 | MR | DOI | Numdam | Zbl

[55] Alexei N. Skorobogatov Torsors and rational points, Cambridge Tracts in Mathematics, 144, Cambridge University Press, 2001, viii+187 pages | DOI | MR | Zbl

[56] Alexei N. Skorobogatov Automorphisms and forms of torus quotients of homogeneous spaces, Mat. Sb., Volume 200 (2009) no. 10, pp. 107-122 | DOI | MR

[57] Bernd Sturmfels; Mauricio Velasco Blow-ups of ℙ n-3 at n points and spinor varieties, J. Commut. Algebra, Volume 2 (2010) no. 2, pp. 223-244 | DOI | MR | Zbl

[58] Geoffrey Timms The nodal cubic surfaces and the surfaces from which they are derived by projection, Proc. R. Soc. Lond., Ser. A, Volume 119 (1928), pp. 213-248 | DOI | Zbl

[59] Pierre Vanhove; Federico Zerbini Single-valued hyperlogarithms, correlation functions and closed string amplitudes, Adv. Theor. Math. Phys., Volume 26 (2022) no. 2, pp. 455-530 | DOI | MR | Zbl

[60] Èrnest B. Vinberg On certain commutative subalgebras of a universal enveloping algebra, Math. USSR, Izv., Volume 36 (1991) no. 1, pp. 1-22 | DOI | Zbl

[61] Zdzisław Wojtkowiak A note on functional equations of the p-adic polylogarithms, Bull. Soc. Math. Fr., Volume 119 (1991) no. 3, pp. 343-370 | MR | DOI | Numdam | Zbl

Cited by Sources: