Abstract
In this paper, we give a complete classification of full exceptional collections, up to cyclic permutations, normalizations and mutations, on smooth toric Fano threefolds and fourfolds with Picard rank two. For such varieties, we find all the exceptional collections of maximal length and show that they are in fact full. This gives a partial answer to a conjecture in [29] and [32]. Moreover, such full exceptional collections essentially arise from Orlov’s theorem on projective bundles.
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Communicated by: D. Plaumann
Acknowledgements
The author would like to express his deep gratitute to his advisor Sung Rak Choi for introducing this topic and for providing many valuable suggestions. He also thanks Professor Alastair Craw for informing us of the paper [35]. Also, he wishes to thank the anonymous referee for careful reading and significant comments. This work waspartially supported by the NationalResearch Foundation of Korea (NRF) grant (No. 2021R1F1A1047563).
References
[1] K. Altmann, J. Buczyński, L. Kastner, A.-L. Winz, Immaculate line bundles on toric varieties. Pure Appl. Math. Q. 16 (2020), 1147–1217. MR4180245 Zbl 1456.1406110.4310/PAMQ.2020.v16.n4.a12Search in Google Scholar
[2] A. Bernardi, S. Tirabassi, Derived categories of toric Fano 3-folds via the Frobenius morphism. Matematiche (Catania) 64 (2009), 117–154. MR2800008 Zbl 1195.14021Search in Google Scholar
[3] A. A. Beĭlinson, Coherent sheaves on ℙn and problems in linear algebra. (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69. English translation: Functional Anal. Appl. 12 (1978), no. 3, 214–216 (1979). MR509388 Zbl 0424.1400310.1007/BF01681436Search in Google Scholar
[4] A. A. Beĭlinson, J. Bernstein, P. Deligne, Faisceaux pervers. In: Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque, 5–171, Soc. Math. France, Paris 1982. MR751966 Zbl 0536.14011Search in Google Scholar
[5] C. Böhning, H.-C. Graf von Bothmer, P. Sosna, On the derived category of the classical Godeaux surface. Adv. Math. 243 (2013), 203–231. MR3062745 Zbl 1299.1401510.1016/j.aim.2013.04.017Search in Google Scholar
[6] C. Böhning, H.-C. Graf von Bothmer, P. Sosna, On the Jordan-Hölder property for geometric derived categories. Adv. Math. 256 (2014), 479–492. MR3177299 Zbl 1311.1402110.1016/j.aim.2014.02.016Search in Google Scholar
[7] A. Bondal, D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences. Compositio Math. 125 (2001), 327–344. MR1818984 Zbl 0994.1800710.1023/A:1002470302976Search in Google Scholar
[8] A. I. Bondal, Representations of associative algebras and coherent sheaves. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 25–44. English translation: Math. USSR-Izv. 34 (1990), 23–42. MR992977 Zbl 0692.1800210.1070/IM1990v034n01ABEH000583Search in Google Scholar
[9] L. Borisov, Z. Hua, On the conjecture of King for smooth toric Deligne-Mumford stacks. Adv. Math. 221 (2009), 277–301. MR2509327 Zbl 1210.1400610.1016/j.aim.2008.11.017Search in Google Scholar
[10] L. Costa, S. Di Rocco, R. M. Miró-Roig, Derived category of fibrations. Math. Res. Lett. 18 (2011), 425–432. MR2802576 Zbl 1253.1401510.4310/MRL.2011.v18.n3.a3Search in Google Scholar
[11] L. Costa, R. M. Miró-Roig, Frobenius splitting and derived category of toric varieties. Illinois J. Math. 54 (2010), 649–669. MR2846477 Zbl 1230.1401810.1215/ijm/1318598676Search in Google Scholar
[12] L. Costa, R. M. Miró-Roig, Derived category of toric varieties with small Picard number. Cent. Eur. J. Math. 10 (2012), 1280–1291. MR2925601 Zbl 1286.1403210.2478/s11533-012-0060-4Search in Google Scholar
[13] A. I. Efimov, Maximal lengths of exceptional collections of line bundles. J. Lond. Math. Soc. (2) 90 (2014), 350–372. MR3263955 Zbl 1318.1404710.1112/jlms/jdu037Search in Google Scholar
[14] S. Galkin, L. Katzarkov, A. Mellit, E. Shinder, Minifolds and phantoms. Preprint 2013, arXiv:1305.4549 [math.AG]Search in Google Scholar
[15] S. Galkin, E. Shinder, Exceptional collections of line bundles on the Beauville surface. Adv. Math. 244 (2013), 1033–1050. MR3077896 Zbl 1408.1406810.1016/j.aim.2013.06.007Search in Google Scholar
[16] W. Hara, Strong full exceptional collections on certain toric varieties with Picard number three via mutations. Matematiche (Catania) 72 (2017), 3–24. MR3731498 Zbl 1402.14022Search in Google Scholar
[17] L. Hille, Exceptional sequences of line bundles on toric varieties. In: Mathematisches Institut, Georg-August-Universität Göttingen: Seminars 2003/2004, 175–190, Universitätsdrucke Göttingen, Göttingen 2004. MR2181579 Zbl 1098.14524Search in Google Scholar
[18] L. Hille, M. Perling, A counterexample to King’s conjecture. Compos. Math. 142 (2006), 1507–1521. MR2278758 Zbl 1108.1404010.1112/S0010437X06002260Search in Google Scholar
[19] L. Hille, M. Perling, Exceptional sequences of invertible sheaves on rational surfaces. Compos. Math. 147 (2011), 1230–1280. MR2822868 Zbl 1237.1404310.1112/S0010437X10005208Search in Google Scholar
[20] A. Hochenegger, N. O. Ilten, Exceptional sequences on rational ℂ∗-surfaces. Manuscripta Math. 142 (2013), 1–34. MR3080998 Zbl 1311.1405110.1007/s00229-012-0591-9Search in Google Scholar
[21] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry. Oxford Univ. Press 2006. MR2244106 Zbl 1095.1400210.1093/acprof:oso/9780199296866.001.0001Search in Google Scholar
[22] M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), 479–508. MR939472 Zbl 0651.1800810.1007/BF01393744Search in Google Scholar
[23] Y. Kawamata, Derived categories of toric varieties. Michigan Math. J. 54 (2006), 517–535. MR2280493 Zbl 1159.1402610.1307/mmj/1163789913Search in Google Scholar
[24] Y. Kawamata, Derived categories of toric varieties II. Michigan Math. J. 62 (2013), 353–363. MR3079267 Zbl 1322.1407510.1307/mmj/1370870376Search in Google Scholar
[25] A. King, Tilting bundles on some rational surfaces. Preprint 1997, http://www.maths.bath.ac.uk/∼masadk/papers/.Search in Google Scholar
[26] P. Kleinschmidt, A classification of toric varieties with few generators. Aequationes Math. 35 (1988), 254–266. MR954243 Zbl 0664.1401810.1007/BF01830946Search in Google Scholar
[27] S. A. Kuleshov, Exceptional and rigid sheaves on surfaces with anticanonical class without base components. J. Math. Sci. 86 (1997), 2951–3003. MR1604186 Zbl 0946.1400610.1007/BF02355112Search in Google Scholar
[28] S. A. Kuleshov, D. O. Orlov, Exceptional sheaves on Del Pezzo surfaces. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 3, 53–87. English translation: Russian Acad. Sci. Izv. Math. 44 (1995), no. 3, 479–513. MR1286839 Zbl 0842.1400910.1070/IM1995v044n03ABEH001609Search in Google Scholar
[29] A. Kuznetsov, Semiorthogonal decompositions in algebraic geometry. In: Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, 635–660, Kyung Moon Sa, Seoul 2014. MR3728631 Zbl 1373.18009Search in Google Scholar
[30] M. Lasoń, M. Michałek, On the full, strongly exceptional collections on toric varieties with Picard number three. Collect. Math. 62 (2011), 275–296. MR2825714 Zbl 1238.1403910.1007/s13348-011-0044-xSearch in Google Scholar
[31] D. W. Lee, Appendix to classification of full exceptional collections on smooth toric Fano varieties with Picard rank two. Preprint 2021, available at https://sites.google.com/yonsei.ac.kr/daewonlee.Search in Google Scholar
[32] W. Liu, S. Yang, X. Yu, Classification of full exceptional collections of line bundles on three blow-ups of ℙ3 J. Korean Math. Soc. 56 (2019), 387–419. MR3922023 Zbl 07094434Search in Google Scholar
[33] M. Michałek, Family of counterexamples to King’s conjecture. C. R. Math. Acad. Sci. Paris 349 (2011), 67–69. MR2755699 Zbl 1211.1405610.1016/j.crma.2010.11.027Search in Google Scholar
[34] D. O. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 852–862. English translation: Russian Acad. Sci. Izv. Math. 41 (1993), 133–141. MR1208153 Zbl 0798.1400710.1070/IM1993v041n01ABEH002182Search in Google Scholar
[35] N. Prabhu-Naik, Tilting bundles on toric Fano fourfolds. J. Algebra 471 (2017), 348–398. MR3569189 Zbl 1386.1401510.1016/j.jalgebra.2016.09.007Search in Google Scholar
[36] H. Uehara, Exceptional collections on toric Fano threefolds and birational geometry. Internat. J. Math. 25 (2014), 1450072, 32 pages. MR3238094 Zbl 1310.1402510.1142/S0129167X14500724Search in Google Scholar
[37] K. Watanabe, M. Watanabe, The classification of Fano 3-folds with torus embeddings. Tokyo J. Math. 5 (1982), 37–48. MR670903 Zbl 0581.1402810.3836/tjm/1270215033Search in Google Scholar
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