Exceptional sequences of line bundles on projective bundles
DOI:
https://doi.org/10.7146/math.scand.a-157081Abstract
For a vector bundle $\mathcal {E}\to \mathbb {P}^\ell $ we investigate exceptional sequences of line bundles on the total space of the projectivisation $X=\mathbb {P}(\mathcal {E})$. In particular, we consider the case of the cotangent bundle of $\mathbb {P}^\ell $. If $\ell =2$, we completely classify the (strong) exceptional sequences and show that any maximal exceptional sequence is full. For general $\ell $, we prove that the Rouquier dimension of $\mathcal {D}(X)$ equals $\dim X$, thereby confirming a conjecture of Orlov.
References
Altmann, K., and Altmann, M., Exceptional sequences of $8$ line bundles on ((mathbb P^1 )^3), J. Algebraic Combin. 56 (2022), no. 2, 305–322. https://doi.org/10.1007/s10801-021-01112-z
Altmann, K., Buczyński, J., Kastner, L., and Winz, A.-L., Immaculate line bundles on toric varieties, Pure Appl. Math. Q. 16 (2020), no. 4, 1147–1217. https://doi.org/10.4310/PAMQ.2020.v16.n4.a12
Altmann, K., and Witt, F., The structure of exceptional sequences on toric varieties of Picard rank two, Algebr. Comb. 7 (2024), no. 4, 1039–1074. https://doi.org/10.5802/alco.371
Beu ılinson, A. A., Coherent sheaves on $mathbb P^n$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69.
Ballard, M., and Favero, D., Hochschild dimensions of tilting objects, Int. Math. Res. Not. IMRN 2012, no. 11, 2607–2645. https://doi.org/10.1093/imrn/rnr124
Böhning, C., Graf von Bothmer, H.-C., Katzarkov, L., and Sosna, P., Determinantal Barlow surfaces and phantom categories, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 7, 1569–1592. https://doi.org/10.4171/JEMS/539
Brion, M., Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, 33–85, Trends Math., Birkhäuser, Basel, 2005. https://doi.org/10.1007/3-7643-7342-3_2
Borisov, L., and Wang, C., On strong exceptional collections of line bundles of maximal length on Fano toric Deligne-Mumford stacks, Asian J. Math. 25 (2021), no. 4, 505–520. https://doi.org/10.4310/AJM.2021.v25.n4.a3
Favero, D., and Huang, J., Rouquier dimension is Krull dimension for normal toric varieties, Eur. J. Math. 9 (2023), no. 4, Paper No. 91, 13 pp. https://doi.org/10.1007/s40879-023-00686-1
Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Vol. 2, Springer-Verlag, Berlin, 1984. https://doi.org/10.1007/978-3-662-02421-8
Fulton, W., Intersection theory, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Vol. 2, Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-1-4612-1700-8
Kleinschmidt, P., A classification of toric varieties with few generators, Aequationes Math. 35 (1988), no. 2–3, 254–266. https://doi.org/10.1007/BF01830946
Krah, J., A phantom on a rational surface, Invent. Math. 235 (2024), no. 3, 1009–1018. https://doi.org/10.1007/s00222-023-01234-0
Kuznetsov, A., Semiorthogonal decompositions in algebraic geometry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, 635–660, Kyung Moon Sa, Seoul, 2014.
Lazarsfeld, R., Positivity in algebraic geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 49, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-642-18808-4
Orlov, D., Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 133–141. https://doi.org/10.1070/IM1993v041n01ABEH002182
Orlov, D., Remarks on generators and dimensions of triangulated categories, Mosc. Math. J. 9 (2009), no. 1, 153–159 https://doi.org/10.17323/1609-4514-2009-9-1-143-149
Rouquier, R., Dimensions of triangulated categories, J. K-Theory 1 (2008), no. 2, 193–256. https://doi.org/10.1017/is007011012jkt010
Rudakov, A., Helices and vector bundles. Seminaire Rudakov. London Mathematical Society Lecture Note Series, 148. Cambridge University Press, Cambridge, 1990.
Solá Conde, L., Geometry of rational homogeneous spaces, Manuscript Łukecin September 9–13, 2019.
