Geodesic
On the MMP for rank one foliations on threefolds
Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e20

Voir la notice de l'article provenant de la source Cambridge University Press

We prove existence of flips for log canonical foliated pairs of rank one on a ${\mathbb Q}$-factorial projective klt threefold. This, in particular, provides a proof of the existence of a minimal model for a rank one foliation on a threefold for a wider range of singularities, after McQuillan. 
Cascini, Paolo; Spicer, Calum. On the MMP for rank one foliations on threefolds. Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e20. doi: 10.1017/fmp.2025.10013
@article{10_1017_fmp_2025_10013,
     author = {Cascini, Paolo and Spicer, Calum},
     title = {On the {MMP} for rank one foliations on threefolds},
     journal = {Forum of Mathematics, Pi},
     pages = {e20},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fmp.2025.10013},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2025.10013/}
}
TY  - JOUR
AU  - Cascini, Paolo
AU  - Spicer, Calum
TI  - On the MMP for rank one foliations on threefolds
JO  - Forum of Mathematics, Pi
PY  - 2025
SP  - e20
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2025.10013/
DO  - 10.1017/fmp.2025.10013
ID  - 10_1017_fmp_2025_10013
ER  - 
%0 Journal Article
%A Cascini, Paolo
%A Spicer, Calum
%T On the MMP for rank one foliations on threefolds
%J Forum of Mathematics, Pi
%D 2025
%P e20
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2025.10013/
%R 10.1017/fmp.2025.10013
%F 10_1017_fmp_2025_10013

[Amb03] Ambro, F., ‘Quasi-log varieties’, Tr. Mat. Inst. Steklova 240(Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry) (2003), 220–239. Google Scholar

[Amb04] Ambro, F., ‘Shokurov’s boundary property’, J. Differential Geom. 67(2) (2004), 229–255.10.4310/jdg/1102536201 Google Scholar | DOI

[Art70] Artin, M., ‘Algebraization of formal moduli: II’, Ann. Math. 91 (1970), 88–135.10.2307/1970602 Google Scholar | DOI

[Bir17] Birkar, C., ‘The augmented base locus of real divisors over arbitrary fields’, Math. Ann. 368(3–4) (2017), 905–921.10.1007/s00208-016-1441-y Google Scholar | DOI

[BM16] Bogomolov, F. and Mcquillan, M., ‘Rational curves on foliated varieties’, in Foliation Theory in Algebraic Geometry, Simons Symp. (Springer, Cham, 2016), 21–51.10.1007/978-3-319-24460-0_2 Google Scholar | DOI

[Bru15] Brunella, M., Birational Geometry of Foliations, vol. 1, IMPA Monographs (Springer, Cham, 2015).10.1007/978-3-319-14310-1 Google Scholar | DOI

[CL77] Carrell, J. B. and Lieberman, D. I., ‘Vector fields and Chern numbers’, Math. Ann. 225(3) (1977), 263–273.10.1007/BF01425242 Google Scholar | DOI

[CN85] Camacho, C. and Neto, A.L., Geometric Theory of Foliations (Birkhäuser Boston, Boston, MA, 1985).10.1007/978-1-4612-5292-4 Google Scholar | DOI

[CS21] Cascini, P. and Spicer, C., ‘MMP for co-rank one foliations on threefolds’, Invent. Math. 225(2) (2021), 603–690.10.1007/s00222-021-01037-1 Google Scholar | DOI

[CS25a] Cascini, P. and Spicer, C., ‘Foliation adjunction’, Math. Ann. 391(4) (2025), 5695–5727.10.1007/s00208-024-03067-5 Google Scholar PubMed | DOI

[CS25b] Cascini, P. and Spicer, C., ‘On base point freeness for rank one foliations’, preprint (2025). Google Scholar

[Dru21] Druel, S., ‘Codimension 1 foliations with numerically trivial canonical class on singular spaces’, Duke Math. J. 170(1) (2021), 95–203.10.1215/00127094-2020-0041 Google Scholar | DOI

[Ful84] Fulton, W., Intersection Theory (Springer-Verlag, 1984).10.1007/978-3-662-02421-8 Google Scholar | DOI

[GKK10] Greb, D., Kebekus, S. and Kovács, S. J., ‘Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties’, Compos. Math. 146(1) (2010), 193–219.10.1112/S0010437X09004321 Google Scholar | DOI

[GKKP11] Greb, D., Kebekus, S., Kovács, S. J. and Peternell, T., ‘Differential forms on log canonical spaces’, Publ. Math. Inst. Hautes Études Sci. 114 (2011), 87–169.10.1007/s10240-011-0036-0 Google Scholar | DOI

[Jou78] Jouanolou, J.-P., ‘Hypersurfaces solutions d’une équation de Pfaff analytique’, Math. Ann. 232(3) (1978), 239–245.10.1007/BF01351428 Google Scholar | DOI

[KM98] Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, vol. 134, Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 1998).10.1017/CBO9780511662560 Google Scholar | DOI

[Kol96] Kollár, J., Rational Curves on Algebraic Varieties, vol. 32, Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer, 1996).10.1007/978-3-662-03276-3 Google Scholar | DOI

[Kol07a] Kollár, J., ‘Kodaira’s canonical bundle formula and adjunction’, in Flips for 3-folds and 4-folds, vol. 35, Oxford Lecture Ser. Math. Appl. (Oxford University Press, Oxford, 2007), 134–162.10.1093/acprof:oso/9780198570615.003.0008 Google Scholar | DOI

[Kol07b] Kollár, J., Lectures on Resolution of Singularities, Annals of Mathematics Studies, vol. 166 (Princeton Univ. Press, Princeton, NJ, 2007). Google Scholar

[Kol13] Kollár, J., Singularities of the Minimal Model Program, vol. 200, Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 2013).10.1017/CBO9781139547895 Google Scholar | DOI

[Mar81] Martinet, J., ‘Normalisation des champs de vecteurs holomorphes (d’après A.-D. Brjuno)’, in Bourbaki Seminar, Vol. 1980/81, vol. 901, Lecture Notes in Math. (Springer, Berlin–New York, 1981), 55–70. Google Scholar

[McQ04] Mcquillan, M., ‘Semi-stable reduction of foliations’, preprint (2004). Google Scholar

[McQ08] Mcquillan, M., ‘Canonical models of foliations’, Pure Appl. Math. Q. 4(3, part 2) (2008), 877–1012.10.4310/PAMQ.2008.v4.n3.a9 Google Scholar | DOI

[Men00] Mendes, L.G., ‘Kodaira dimension of holomorphic singular foliations’, Bol. Soc. Brasil. Mat. (N.S.) 31(2) (2000), 127–143.10.1007/BF01244239 Google Scholar | DOI

[MP13] Mcquillan, M. and Panazzolo, D., ‘Almost étale resolution of foliations’, J. Differential Geom. 95(2) (2013), 279–319.10.4310/jdg/1376053448 Google Scholar | DOI

[Sei67] Seidenberg, A., ‘Differential ideals in rings of finitely generated type’, Amer. J. Math. 89 (1967), 22–42.10.2307/2373093 Google Scholar | DOI

[Spi20] Spicer, C., ‘Higher-dimensional foliated Mori theory’, Compos. Math. 156(1) (2020), 1–38.10.1112/S0010437X19007681 Google Scholar | DOI

[SS22] Spicer, C. and Svaldi, R., ‘Local and global applications of the Minimal Model Program for co-rank 1 foliations on threefolds’, J. Eur. Math. Soc. (JEMS) 24(11) (2022), 3969–4025.10.4171/jems/1173 Google Scholar | DOI

Cité par Sources :