Non-existence of phantoms on some non-generic blowups of the projective plane

Borisov, Lev; Kemboi, Kimoi

doi:10.1090/proc/17105

Non-existence of phantoms on some non-generic blowups of the projective plane
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by Lev Borisov and Kimoi Kemboi;

Proc. Amer. Math. Soc. 153 (2025), 963-968

DOI: https://doi.org/10.1090/proc/17105

Published electronically: January 29, 2025

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Abstract:

We show that blowups of the projective plane at points lying on a smooth cubic curve do not contain phantoms, provided the points are chosen in very general position on this curve.

References

Nicolas Addington, New derived symmetries of some hyperkähler varieties, Algebr. Geom. 3 (2016), no. 2, 223–260. MR 3477955, DOI 10.14231/AG-2016-011
Rina Anno and Timothy Logvinenko, Spherical DG-functors, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 9, 2577–2656. MR 3692883, DOI 10.4171/JEMS/724
Igor Burban and Bernd Kreußler, Derived categories of irreducible projective curves of arithmetic genus one, Compos. Math. 142 (2006), no. 5, 1231–1262. MR 2264663, DOI 10.1112/S0010437X06002090
Christian Böhning, Hans-Christian Graf von Bothmer, Ludmil Katzarkov, and Pawel Sosna, Determinantal Barlow surfaces and phantom categories, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 7, 1569–1592. MR 3361723, DOI 10.4171/JEMS/539
Christian Böhning, Hans-Christian Graf von Bothmer, and Pawel Sosna, On the derived category of the classical Godeaux surface, Adv. Math. 243 (2013), 203–231. MR 3062745, DOI 10.1016/j.aim.2013.04.017
C. Diemer, L. Katzarkov, and G. Kerr, Compactifications of spaces of Landau-Ginzburg models, Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), no. 3, 55–76; English transl., Izv. Math. 77 (2013), no. 3, 487–508. MR 3098787, DOI 10.4213/im8019
Alexander Efimov, Wall finiteness obstruction for dg categories and for algebras over colored dg operads, Talk Slides (2020).
Sergey Gorchinskiy and Dmitri Orlov, Geometric phantom categories, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 329–349. MR 3090263, DOI 10.1007/s10240-013-0050-5
Lutz Hille and Michel Van den Bergh, Fourier-Mukai transforms, Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332, Cambridge Univ. Press, Cambridge, 2007, pp. 147–177. MR 2384610, DOI 10.1017/CBO9780511735134.007
Johannes Krah, A phantom on a rational surface, Invent. Math. 235 (2024), no. 3, 1009–1018. MR 4701883, DOI 10.1007/s00222-023-01234-0
Alexander Kuznetsov, Base change for semiorthogonal decompositions, Compos. Math. 147 (2011), no. 3, 852–876. MR 2801403, DOI 10.1112/S0010437X10005166
Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 635–660. MR 3728631
Dmitrii Pirozhkov, Admissible subcategories of del Pezzo surfaces, Adv. Math. 424 (2023), Paper No. 109046, 62. MR 4581971, DOI 10.1016/j.aim.2023.109046
Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, DOI 10.1215/S0012-7094-01-10812-0
Bertrand Toën, The homotopy theory of $dg$-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615–667. MR 2276263, DOI 10.1007/s00222-006-0025-y