On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors  [ Faisceaux de Verlinde et dualité étrange pour les diviseurs de Noether-Lefschetz elliptiques ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 5, p. 2067-2086
On établit l’isomorphisme de dualité étrange pour toutes les surfaces K3 constituant un diviseur de Noether-Lefschetz dans l’espace de modules de surfaces K3 quasipolarisées. On interprète le résultat d’une manière globale, comme un isomorphisme de faisceaux à travers ce diviseur, et on décrit aussi la construction globale sur l’espace de modules des surfaces K3s polarisées.
We extend results on generic strange duality for K3 surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized K3s. We interpret the statement globally as an isomorphism of sheaves over this divisor, and also describe the global construction over the space of polarized K3s.
DOI : https://doi.org/10.5802/aif.2904
Classification:  14J60,  14J28,  14J15
Mots clés: surface K3, espace de modules des faisceaux, dualité étrange
@article{AIF_2014__64_5_2067_0,
     author = {Marian, Alina and Oprea, Dragos},
     title = {On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {5},
     year = {2014},
     pages = {2067-2086},
     doi = {10.5802/aif.2904},
     mrnumber = {3330931},
     zbl = {06387331},
     language = {en},
     url = {http://http://www.numdam.org/item/AIF_2014__64_5_2067_0}
}
Marian, Alina; Oprea, Dragos. On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors. Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2067-2086. doi : 10.5802/aif.2904. http://www.numdam.org/item/AIF_2014__64_5_2067_0/

[1] Barth, W.; Peters, C.; Van De Ven, A. Compact complex surfaces, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 4 (1984), pp. x+304 | Article | MR 749574 | Zbl 1036.14016

[2] Bernardara, M.; Hein, G. The Euclid-Fourier-Mukai algorithm for elliptic surfaces (arXiv:1002.4986, to appear in Asian J. Math.)

[3] Bridgeland, Tom Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math., Tome 498 (1998), pp. 115-133 | Article | MR 1629929 | Zbl 0905.14020

[4] Caldararu, Andrei Horia Derived categories of twisted sheaves on Calabi-Yau manifolds, ProQuest LLC, Ann Arbor, MI (2000), pp. 196 (Thesis (Ph.D.)–Cornell University) | MR 2700538

[5] Ellingsrud, Geir; Göttsche, Lothar; Lehn, Manfred On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom., Tome 10 (2001) no. 1, pp. 81-100 | MR 1795551 | Zbl 0976.14002

[6] Friedman, Robert Algebraic surfaces and holomorphic vector bundles, Springer-Verlag, New York, Universitext (1998), pp. x+328 | Article | MR 1600388 | Zbl 0902.14029

[7] Van Der Geer, Gerard; Katsura, Toshiyuki Note on tautological classes of moduli of K3 surfaces, Mosc. Math. J., Tome 5 (2005) no. 4, p. 775-779, 972 | MR 2266459 | Zbl 1124.14011

[8] Hernandez Ruiperez, D.; Lopez Martin, A. C.; Sancho De Salas, F. Relative integral functors for singular fibrations and singular partners, J. Eur. Math. Soc., Tome 11 (2009), pp. 597-625 | Article | MR 2505443 | Zbl 1221.18010

[9] Huybrechts, Daniel Birational symplectic manifolds and their deformations, J. Differential Geom., Tome 45 (1997) no. 3, pp. 488-513 | MR 1472886 | Zbl 0917.53010

[10] Le Potier, J. Fibré déterminant et courbes de saut sur les surfaces algébriques, Complex projective geometry (Trieste, 1989/Bergen, 1989), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 179 (1992), pp. 213-240 | Article | MR 1201385 | Zbl 0788.14045

[11] Le Potier, J. Dualité étrange sur le plan projectif (1996) (Luminy)

[12] Li, Jun Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom., Tome 37 (1993) no. 2, pp. 417-466 | MR 1205451 | Zbl 0809.14006

[13] Marian, Alina; Oprea, Dragos A tour of theta dualities on moduli spaces of sheaves, Curves and abelian varieties, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 465 (2008), pp. 175-201 | Article | MR 2457738 | Zbl 1149.14301

[14] Marian, Alina; Oprea, Dragos Generic strange duality for K3 surfaces, Duke Math. J., Tome 162 (2013) no. 8, pp. 1463-1501 (With an appendix by Kota Yoshioka) | Article | MR 3079253 | Zbl 1275.14037

[15] Piatetski-Shapiro, I. I.; Shafarevich, I. R. A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izvestia, Tome 5 (1971), pp. 547-588 | Article | Zbl 0253.14006

[16] Sawon, Justin Abelian fibred holomorphic symplectic manifolds, Turkish J. Math., Tome 27 (2003) no. 1, pp. 197-230 | MR 1975339 | Zbl 1065.53067

[17] Scala, Luca Dualité étrange de Le Potier et cohomologie du schéma de Hilbert ponctuel d’une surface, Gaz. Math. (2007) no. 112, pp. 53-65 | MR 2319915 | Zbl 1120.14032

[18] Scala, Luca Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles, Duke Math. J., Tome 150 (2009) no. 2, pp. 211-267 | Article | MR 2569613 | Zbl 1211.14012