×

Triple planes with \(p_g=q=0\). (English) Zbl 1422.14024

A triple plane is a finite flat morphism \(f : X \rightarrow \mathbb{P}^2\). In this paper, the authors show that smooth projective surfaces \(X\) over \(\mathbb{C}\) which arise as general triple planes and such that \(p_g(X)=q(X)=0\) belong to at most 12 families which the authors name I, II,\(\ldots\), XII. The authors show that the families of type I to VII exist and they completely classify those of type I to VI.

MSC:

14E20 Coverings in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Software:

Macaulay2
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alexander, James, Surfaces rationnelles non-sp\'eciales dans \({\bf P}^4\), Math. Z., 200, 1, 87-110 (1988) · Zbl 0702.14031
[2] Ancona, Vincenzo; Ottaviani, Giorgio, Unstable hyperplanes for Steiner bundles and multidimensional matrices, Adv. Geom., 1, 2, 165-192 (2001) · Zbl 0983.14034
[3] Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J., Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 267, xvi+386 pp. (1985), Springer-Verlag, New York · Zbl 0559.14017
[4] Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 4, xii+436 pp. (2004), Springer-Verlag, Berlin · Zbl 1036.14016
[5] Beltrametti, Mauro C.; Sommese, Andrew J., The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics 16, xxii+398 pp. (1995), Walter de Gruyter & Co., Berlin · Zbl 0845.14003
[6] Bronowski, J., On triple planes. II, J. London Math. Soc., 17, 24-31 (1942) · Zbl 0063.00612
[7] Cascini, Paolo, On the moduli space of the Schwarzenberger bundles, Pacific J. Math., 205, 2, 311-323 (2002) · Zbl 1063.14012
[8] Casnati, G.; Ekedahl, T., Covers of algebraic varieties. I. A general structure theorem, covers of degree \(3,4\) and Enriques surfaces, J. Algebraic Geom., 5, 3, 439-460 (1996) · Zbl 0866.14009
[9] F. Conforto, Le superficie razionali, Zanichelli, Bologna, 1945, pp. xv+554.
[10] Conte, A.; Verra, A., Reye constructions for nodal Enriques surfaces, Trans. Amer. Math. Soc., 336, 1, 79-100 (1993) · Zbl 0849.14015
[11] Cossec, Fran\c{c}ois R., Reye congruences, Trans. Amer. Math. Soc., 280, 2, 737-751 (1983) · Zbl 0541.14034
[12] Decker, Wolfram; Ein, Lawrence; Schreyer, Frank-Olaf, Construction of surfaces in \({\bf P}_4\), J. Algebraic Geom., 2, 2, 185-237 (1993) · Zbl 0795.14019
[13] Dimca, Alexandru, Singularities and topology of hypersurfaces, Universitext, xvi+263 pp. (1992), Springer-Verlag, New York · Zbl 0753.57001
[14] Dolgachev, Igor V., Classical algebraic geometry, xii+639 pp. (2012), Cambridge University Press, Cambridge · Zbl 1252.14001
[15] Dolgachev, I.; Kapranov, M., Arrangements of hyperplanes and vector bundles on \(\mathbf{P}^n\), Duke Math. J., 71, 3, 633-664 (1993) · Zbl 0804.14007
[16] Val, Patrick du, On triple planes having branch curves of order not greater than twelve, J. London Math. Soc., 8, 3, 199-206 (1933) · Zbl 0007.17405
[17] du Val, Patrick, On triple planes whose branch curves are of order fourteen, Proc. London Math. Soc. (2), 39, 1, 68-81 (1935) · Zbl 0011.07803
[18] Faenzi, Daniele; Matei, Daniel; Vall\`es, Jean, Hyperplane arrangements of Torelli type, Compos. Math., 149, 2, 309-332 (2013) · Zbl 1278.14027
[19] Faenzi, Daniele; Stipins, Janis, A small resolution for triple covers in algebraic geometry, Matematiche (Catania), 56, 2, 257-267 (2003) (2001) · Zbl 1051.14016
[20] Faenzi, Daniele; Vall\`es, Jean, Logarithmic bundles and line arrangements, an approach via the standard construction, J. London Math. Soc. (2), 90, 3, 675-694 (2014) · Zbl 1308.52021
[21] Fujita, Takao, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series 155, xiv+205 pp. (1990), Cambridge University Press, Cambridge · Zbl 0743.14004
[22] Fulton, William; Harris, Joe, Representation theory, Graduate Texts in Mathematics, Readings in Mathematics 129, xvi+551 pp. (1991), Springer-Verlag, New York · Zbl 0744.22001
[23] D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
[24] Gross, Mark, Surfaces of bidegree \((3,n)\) in \({\rm Gr}(1,{\bf P}^3)\), Math. Z., 212, 1, 73-106 (1993) · Zbl 0812.14033
[25] Grothendieck, Alexander, Rev\^etements \'etales et groupe fondamental. Fasc. I: Expos\'es 1 \`“a 5, S\'”eminaire de G\'eom\'etrie Alg\'ebrique 1960/61, iv+143 pp. (not consecutively paged) (loose errata) pp. (1963), Institut des Hautes \'Etudes Scientifiques, Paris
[26] Harris, Joe, Algebraic geometry, Graduate Texts in Mathematics 133, xx+328 pp. (1992), Springer-Verlag, New York · Zbl 0779.14001
[27] Hartshorne, Robin, Algebraic geometry, Graduate Texts in Mathematics 52, xvi+496 pp. (1977), Springer-Verlag, New York-Heidelberg · Zbl 0367.14001
[28] Hartshorne, Robin, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, vii+423 pp. (1966), Springer-Verlag, Berlin-New York · Zbl 0212.26101
[29] Hoppe, Hans J\`“urgen, Generischer Spaltungstyp und zweite Chernklasse stabiler Vektorraumb\'”undel vom Rang \(4\)auf \({\bf P}_4\), Math. Z., 187, 3, 345-360 (1984) · Zbl 0567.14011
[30] Huybrechts, Daniel; Lehn, Manfred, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, xiv+269 pp. (1997), Friedr. Vieweg & Sohn, Braunschweig · Zbl 0872.14002
[31] Ionescu, Paltin, Embedded projective varieties of small invariants. Algebraic geometry, Bucharest 1982, Bucharest, 1982, Lecture Notes in Math. 1056, 142-186 (1984), Springer, Berlin
[32] Ionescu, Paltin, Vari\'et\'es projectives lisses de degr\'es \(5\)et \(6\), C. R. Acad. Sci. Paris S\'er. I Math., 293, 15, 685-687 (1981) · Zbl 0516.14025
[33] Iversen, Birger, Numerical invariants and multiple planes, Amer. J. Math., 92, 968-996 (1970) · Zbl 0232.14013
[34] Lanteri, Antonio; Palleschi, Marino, About the adjunction process for polarized algebraic surfaces, J. Reine Angew. Math., 352, 15-23 (1984) · Zbl 0535.14003
[35] Le Barz, Patrick, Quelques formules multis\'ecantes pour les surfaces. Enumerative geometry, Sitges, 1987, Lecture Notes in Math. 1436, 151-188 (1990), Springer, Berlin
[36] Miranda, Rick, Triple covers in algebraic geometry, Amer. J. Math., 107, 5, 1123-1158 (1985) · Zbl 0611.14011
[37] Okonek, Christian, \"Uber \(2\)-codimensionale Untermannigfaltigkeiten vom Grad \(7\)in \({\bf P}^4\)und \({\bf P}^5\), Math. Z., 187, 2, 209-219 (1984) · Zbl 0575.14030
[38] Okonek, Christian; Schneider, Michael; Spindler, Heinz, Vector bundles on complex projective spaces, Progress in Mathematics 3, vii+389 pp. (1980), Birkh\"auser, Boston, Mass. · Zbl 0438.32016
[39] G. Ottaviani, Variet\`“a proiettive di codimensione piccola, Vol. 22, Quaderni dell”Indam [Publications of the Istituto di Alta Matematica Francesco Severi], Aracne Editrice, 1995, p. 96.
[40] Pardini, R., Triple covers in positive characteristic, Ark. Mat., 27, 2, 319-341 (1989) · Zbl 0707.14010
[41] Seifert, Herbert; Threlfall, William, Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics 89, xvi+437 pp. (1980), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London
[42] Sommese, Andrew John, Hyperplane sections of projective surfaces. I. The adjunction mapping, Duke Math. J., 46, 2, 377-401 (1979) · Zbl 0415.14019
[43] Sommese, Andrew John; Van de Ven, A., On the adjunction mapping, Math. Ann., 278, 1-4, 593-603 (1987) · Zbl 0655.14001
[44] Tan, Sheng-Li, Triple covers on smooth algebraic varieties. Geometry and nonlinear partial differential equations, Hangzhou, 2001, AMS/IP Stud. Adv. Math. 29, 143-164 (2002), Amer. Math. Soc., Providence, RI · Zbl 1018.14003
[45] Vall\`“es, Jean, Fibr\'”es de Schwarzenberger et coniques de droites sauteuses, Bull. Soc. Math. France, 128, 3, 433-449 (2000) · Zbl 0955.14009
[46] Vall\`“es, Jean, Nombre maximal d”hyperplans instables pour un fibr\'e de Steiner, Math. Z., 233, 3, 507-514 (2000) · Zbl 0952.14011
[47] Weyman, Jerzy, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics 149, xiv+371 pp. (2003), Cambridge University Press, Cambridge · Zbl 1075.13007
[48] Zariski, Oscar, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math., 51, 2, 305-328 (1929) · JFM 55.0806.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.