On the genus of reducible surfaces and degenerations of surfaces

Alberto Calabri[1]; Ciro Ciliberto[2]; Flaminio Flamini[3]; Rick Miranda[4]

  • [1] Università degli Studi di Padova Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Via Trieste, 63 35121 Padova (Italy)
  • [2] Università degli Studi di Roma “Tor Vergata" Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italy)
  • [3] Università degli Studi di Roma “Tor Vergadata” Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italy)
  • [4] Colorado State University Department of Mathematics 101 Weber Building Fort Collins, CO 80523–1874 (USA)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 2, page 491-516
  • ISSN: 0373-0956

Abstract

top
We deal with a reducible projective surface X with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the ω -genus p ω ( X ) of X , i.e. the dimension of the vector space of global sections of the dualizing sheaf ω X . Then we prove that, when X is smoothable, i.e. when X is the central fibre of a flat family π : 𝒳 → Δ parametrized by a disc, with smooth general fibre, then the ω -genus of the fibres of π is constant.

How to cite

top

Calabri, Alberto, et al. "On the genus of reducible surfaces and degenerations of surfaces." Annales de l’institut Fourier 57.2 (2007): 491-516. <http://eudml.org/doc/10230>.

@article{Calabri2007,
abstract = {We deal with a reducible projective surface $X$ with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the $\omega $-genus$p_\omega (X)$ of $X$, i.e. the dimension of the vector space of global sections of the dualizing sheaf $\omega _X$. Then we prove that, when $X$ is smoothable, i.e. when $X$ is the central fibre of a flat family $\pi :\mathcal\{X\}\rightarrow \Delta $ parametrized by a disc, with smooth general fibre, then the $\omega $-genus of the fibres of $\pi $ is constant.},
affiliation = {Università degli Studi di Padova Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Via Trieste, 63 35121 Padova (Italy); Università degli Studi di Roma “Tor Vergata" Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italy); Università degli Studi di Roma “Tor Vergadata” Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italy); Colorado State University Department of Mathematics 101 Weber Building Fort Collins, CO 80523–1874 (USA)},
author = {Calabri, Alberto, Ciliberto, Ciro, Flamini, Flaminio, Miranda, Rick},
journal = {Annales de l’institut Fourier},
keywords = {Degenerations of surfaces; singularities; birational geometry; topological invariants; degeneration of surfaces},
language = {eng},
number = {2},
pages = {491-516},
publisher = {Association des Annales de l’institut Fourier},
title = {On the genus of reducible surfaces and degenerations of surfaces},
url = {http://eudml.org/doc/10230},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Calabri, Alberto
AU - Ciliberto, Ciro
AU - Flamini, Flaminio
AU - Miranda, Rick
TI - On the genus of reducible surfaces and degenerations of surfaces
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 2
SP - 491
EP - 516
AB - We deal with a reducible projective surface $X$ with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the $\omega $-genus$p_\omega (X)$ of $X$, i.e. the dimension of the vector space of global sections of the dualizing sheaf $\omega _X$. Then we prove that, when $X$ is smoothable, i.e. when $X$ is the central fibre of a flat family $\pi :\mathcal{X}\rightarrow \Delta $ parametrized by a disc, with smooth general fibre, then the $\omega $-genus of the fibres of $\pi $ is constant.
LA - eng
KW - Degenerations of surfaces; singularities; birational geometry; topological invariants; degeneration of surfaces
UR - http://eudml.org/doc/10230
ER -

References

top
  1. Alberto Calabri, Ciro Ciliberto, Flaminio Flamini, Rick Miranda, On the K 2 of degenerations of surfaces and the multiple point formula Zbl1125.14018
  2. Alberto Calabri, Ciro Ciliberto, Flaminio Flamini, Rick Miranda, On the geometric genus of reducible surfaces and degenerations of surfaces to unions of planes, The Fano Conference (2004), 277-312, Univ. Torino, Turin Zbl1071.14057MR2112579
  3. Ciro Ciliberto, Angelo Lopez, Rick Miranda, Projective degenerations of K 3 surfaces, Gaussian maps, and Fano threefolds, Invent. Math. 114 (1993), 641-667 Zbl0807.14028MR1244915
  4. Ciro Ciliberto, Rick Miranda, Mina Teicher, Pillow degenerations of K 3 surfaces, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) 36 (2001), 53-63, Kluwer Acad. Publ., Dordrecht Zbl1006.14014MR1866890
  5. Daniel C. Cohen, Alexander I. Suciu, The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv. 72 (1997), 285-315 Zbl0959.52018MR1470093
  6. The birational geometry of degenerations, 29 (1983), FriedmanRobertR., Mass. 
  7. William Fulton, Introduction to toric varieties, 131 (1993), Princeton University Press, Princeton, NJ Zbl0813.14039MR1234037
  8. Robin Hartshorne, Families of curves in ℙ 3 and Zeuthen’s problem, Mem. Amer. Math. Soc. 130 (1997) Zbl0894.14001MR1401493
  9. G. Kempf, Finn Faye Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings. I, (1973), Springer-Verlag, Berlin Zbl0271.14017MR335518
  10. János Kollár, Toward moduli of singular varieties, Compositio Math. 56 (1985), 369-398 Zbl0666.14003MR814554
  11. B. G. Moishezon, Stable branch curves and braid monodromies, Algebraic geometry (Chicago, Ill., 1980) 862 (1981), 107-192, Springer, Berlin Zbl0476.14005MR644819
  12. Boris Moishezon, Mina Teicher, Braid group techniques in complex geometry. III. Projective degeneration of V 3 , Classification of algebraic varieties (L’Aquila, 1992) 162 (1994), 313-332, Amer. Math. Soc., Providence, RI Zbl0815.14023MR1272706
  13. David R. Morrison, The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) 106 (1984), 101-119, Princeton Univ. Press, Princeton, NJ Zbl0576.32034MR756848
  14. Francesco Severi, Vorlesungen über algebraische Geometrie, 1 (1921), Teubner, Leipzig Zbl48.0687.01
  15. M. Teicher, Hirzebruch surfaces: degenerations, related braid monodromy, Galois covers, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) 241 (1999), 305-325, Amer. Math. Soc., Providence, RI Zbl0993.14017MR1720873
  16. Guido Zappa, Su alcuni contributi alla conosceuza della struttura topologica delle superficie algebriche, dati dal metodo dello spezzamento in sistemi di piani, Pont. Acad. Sci. Acta 7 (1943), 4-8 Zbl0061.34908MR26362
  17. Guido Zappa, Alla ricerca di nuovi significati topologici dei generi geometrico e aritmetico di una superficie algebrica, Ann. Mat. Pura Appl. (4) 30 (1949), 123-146 Zbl0041.48006MR36545

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.