Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2018: 0.53

See all formats and pricing
More options …
Volume 19, Issue 4


A Batyrev type classification of ℚ-factorial projective toric varieties

Michele Rossi / Lea Terracini
Published Online: 2018-04-06 | DOI: https://doi.org/10.1515/advgeom-2018-0007


The present paper generalizes, inside the class of projective toric varieties, the classification [2], performed by Batyrev in 1991 for smooth complete toric varieties, to the singular ℚ-factorial case.

Keywords: ℚ-factorial complete toric variety; projective toric bundle; secondary fan; Gale duality; fan and weight matrices; toric cover; splitting fan; primitive collection and relation

MSC 2010: 14M25; 52B20; 52B35


  • [1]

    V. Alexeev, R. Pardini, On the existence of ramified abelian covers. Rend. Semin. Mat. Univ. Politec. Torino 71 (2013), 307–315. MR3506389 Zbl 1327.14074Google Scholar

  • [2]

    V. V. Batyrev, On the classification of smooth projective toric varieties. Tohoku Math. J. (2) 43 (1991), 569–585. MR1133869 Zbl 0792.14026CrossrefGoogle Scholar

  • [3]

    V. V. Batyrev, D. A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties. Duke Math. J. 75 (1994), 293–338. MR1290195 Zbl 0851.14021Google Scholar

  • [4]

    F. Berchtold, J. Hausen, Bunches of cones in the divisor class group—a new combinatorial language for toric varieties. Int. Math. Res. Not. no. 6 (2004), 261–302. MR2041065 Zbl 1137.14313Google Scholar

  • [5]

    C. Casagrande, Contractible classes in toric varieties. Math. Z. 243 (2003), 99–126. MR1953051 Zbl 1077.14070CrossrefGoogle Scholar

  • [6]

    C. Casagrande, On the birational geometry of Fano 4-folds. Math. Ann. 355 (2013), 585–628. MR3010140 Zbl 1260.14052Web of ScienceCrossrefGoogle Scholar

  • [7]

    H. Conrads, Weighted projective spaces and reflexive simplices. Manuscripta Math. 107 (2002), 215–227. MR1894741 Zbl 1013.52009CrossrefGoogle Scholar

  • [8]

    D. A. Cox, The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4 (1995), 17–50. MR1299003 Zbl 0846.14032Google Scholar

  • [9]

    D. A. Cox, J. B. Little, H. K. Schenck, Toric varieties, volume 124 of Graduate Studies in Mathematics. Amer. Math. Soc. 2011. MR2810322 Zbl 1223.14001Google Scholar

  • [10]

    D. A. Cox, C. von Renesse, Primitive collections and toric varieties. Tohoku Math. J. (2) 61 (2009), 309–332. MR2568257 Zbl 1185.14045CrossrefGoogle Scholar

  • [11]

    O. Fujino, S. Payne, Smooth complete toric threefolds with no nontrivial nef line bundles. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), 174–179 (2006). MR2196723 Zbl 1141.14313CrossrefGoogle Scholar

  • [12]

    O. Fujino, H. Sato, Smooth projective toric varieties whose nontrivial nef line bundles are big. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), 89–94. MR2548019 Zbl 1189.14056CrossrefWeb of ScienceGoogle Scholar

  • [13]

    I. M. Gel’ fand, M. M. Kapranov, A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants. Birkhäuser 1994. MR1264417 Zbl 0827.14036Google Scholar

  • [14]

    I. M. Gel’ fand, A. V. Zelevinsky, M. M. Kapranov, Newton polyhedra of principal A-determinants. Dokl. Akad. Nauk SSSR 308 (1989), 20–23. MR1020882 Zbl 0742.14042Google Scholar

  • [15]

    Y. Hu, S. Keel, Mori dream spaces and GIT. Michigan Math. J. 48 (2000), 331–348. MR1786494 Zbl 1077.14554CrossrefGoogle Scholar

  • [16]

    P. Kleinschmidt, A classification of toric varieties with few generators. Aequationes Math. 35 (1988), 254–266. MR954243 Zbl 0664.14018CrossrefGoogle Scholar

  • [17]

    P. Kleinschmidt, B. Sturmfels, Smooth toric varieties with small Picard number are projective. Topology 30 (1991), 289–299. MR1098923 Zbl 0739.14032CrossrefGoogle Scholar

  • [18]

    T. Oda, Torus embeddings and applications, volume 57 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Springer 1978. MR546291 Zbl 0417.14043Google Scholar

  • [19]

    T. Oda, H. S. Park, Linear Gale transforms and Gel’fand-Kapranov-Zelevinskij decompositions. Tohoku Math. J. (2) 43 (1991), 375–399. MR1117211 Zbl 0782.52006CrossrefGoogle Scholar

  • [20]

    M. Reid, Decomposition of toric morphisms. In: Arithmetic and geometry, Vol. II, 395–418, Birkhäuser 1983. MR717617 Zbl 0571.14020Google Scholar

  • [21]

    M. Rossi, L. Terracini, Linear algebra and toric data of weighted projective spaces. Rend. Semin. Mat. Univ. Politec. Torino 70 (2012), 469–495. MR3305560 Zbl 06382843Google Scholar

  • [22]

    M. Rossi, L. Terracini, ℤ-linear Gale duality and poly weighted spaces (PWS). Linear Algebra Appl. 495 (2016), 256–288. MR3462999 Zbl 1332.14065CrossrefWeb of ScienceGoogle Scholar

  • [23]

    M. Rossi, L. Terracini, A ℚ-factorial complete toric variety is a quotient of a poly weighted space. Ann. Mat. Pura Appl. (4) 196 (2017), 325–347. MR3600869 Zbl 06686404CrossrefWeb of ScienceGoogle Scholar

  • [24]

    M. Rossi, L. Terracini, A ℚ-factorial complete toric variety of Picard number 2 is projective. To appear in J. Pure Appl. Algebra. Preprint arXiv:1504.03850 [math.AG]Google Scholar

  • [25]

    M. Rossi, L. Terracini, Fibration and classification of a smooth projective toric variety of low Picard number. To appear in Ann. Mat. Pura Appl. Preprint arXiv:1507.00493 [math.AG]Google Scholar

  • [26]

    M. Rossi, L. Terracini, Erratum to: A-factorial complete toric variety is a quotient of a poly weighted space. Preprint arXiv:1502.00879v3 [math.AG]Google Scholar

  • [27]

    H. Sato, Toward the classification of higher-dimensional toric Fano varieties. Tohoku Math. J. (2) 52 (2000), 383–413. MR1772804 Zbl 1028.14015CrossrefGoogle Scholar

About the article

Received: 2016-04-27

Revised: 2017-09-05

Published Online: 2018-04-06

Published in Print: 2019-10-25

Funding: The authors were partially supported by the MIUR-PRIN 2010-11 Research Funds “Geometria delle Varietà Algebriche”. The first author is also supported by the I.N.D.A.M. as a member of the G.N.S.A.G.A.

Communicated by: M. Joswig

Citation Information: Advances in Geometry, Volume 19, Issue 4, Pages 433–476, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0007.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in