On the projective genus of surfaces

Pietro Sabatino

Bollettino dell'Unione Matematica Italiana (2006)

  • Volume: 9-B, Issue: 2, page 311-317
  • ISSN: 0392-4041

Abstract

top
Let X N be a smooth irreducible non degenerate surface over the complex numbers, N 4 . We define the projective genus of X , denoted by P G ( X ) , as the geometric genus of the singular curve of the projection of X from a general linear subspace of codimension four. Denote by g ( X ) the sectional genus of X . In this paper we conjecture that the only surfaces for which P G ( X ) = g ( X ) - 1 are the del Pezzo surface in 4 , in 5 and a conic bundle of degree 5 in 4 . We prove that for N 5 if P G ( X ) = g ( X ) - 1 + λ , λ a non negative integer, then g ( X ) λ + 1 + α where α = - 2 for a scroll and α = 0 otherwise, and deduce the conjecture for N 5 from this statement.

How to cite

top

Sabatino, Pietro. "On the projective genus of surfaces." Bollettino dell'Unione Matematica Italiana 9-B.2 (2006): 311-317. <http://eudml.org/doc/289631>.

@article{Sabatino2006,
abstract = {Let $X \subset \mathbb\{P\}^N$ be a smooth irreducible non degenerate surface over the complex numbers, $N \geq 4$. We define the projective genus of $X$, denoted by $PG(X)$, as the geometric genus of the singular curve of the projection of $X$ from a general linear subspace of codimension four. Denote by $g(X)$ the sectional genus of $X$. In this paper we conjecture that the only surfaces for which $PG(X) = g(X) - 1$ are the del Pezzo surface in $\mathbb\{P\}^4$, in $\mathbb\{P\}^5$ and a conic bundle of degree 5 in $\mathbb\{P\}^4$. We prove that for $N \geq 5$ if $PG(X) = g(X) - 1 + \lambda$, $\lambda$ a non negative integer, then $g(X) \leq \lambda + 1 + \alpha$ where $\alpha = -2$ for a scroll and $\alpha = 0$ otherwise, and deduce the conjecture for $N \geq 5$ from this statement.},
author = {Sabatino, Pietro},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {311-317},
publisher = {Unione Matematica Italiana},
title = {On the projective genus of surfaces},
url = {http://eudml.org/doc/289631},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Sabatino, Pietro
TI - On the projective genus of surfaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/6//
PB - Unione Matematica Italiana
VL - 9-B
IS - 2
SP - 311
EP - 317
AB - Let $X \subset \mathbb{P}^N$ be a smooth irreducible non degenerate surface over the complex numbers, $N \geq 4$. We define the projective genus of $X$, denoted by $PG(X)$, as the geometric genus of the singular curve of the projection of $X$ from a general linear subspace of codimension four. Denote by $g(X)$ the sectional genus of $X$. In this paper we conjecture that the only surfaces for which $PG(X) = g(X) - 1$ are the del Pezzo surface in $\mathbb{P}^4$, in $\mathbb{P}^5$ and a conic bundle of degree 5 in $\mathbb{P}^4$. We prove that for $N \geq 5$ if $PG(X) = g(X) - 1 + \lambda$, $\lambda$ a non negative integer, then $g(X) \leq \lambda + 1 + \alpha$ where $\alpha = -2$ for a scroll and $\alpha = 0$ otherwise, and deduce the conjecture for $N \geq 5$ from this statement.
LA - eng
UR - http://eudml.org/doc/289631
ER -

References

top
  1. HARTSHORNE, R., Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer (1997). 
  2. BELTRAMETTI, M. - SOMMESE, A. J., The Adjunction Theory of Complex Projective Varieties, De Gruyter expositions in mathematics16, de Gruyter (1995). 
  3. FRANCHETTA, A., Sulla curva doppia della proiezione della superficie generale dell’ S 4 , da un punto generico su un S 3 , Rend. Accad. d’Italia, VII-2 (1941), 282-288. 
  4. FRANCHETTA, A., Sulla curva doppia della proiezione della superficie generale dell’ S 4 , da un punto generico su un S 3 , Rend. Accad. Naz. Lincei, VIII-2 (1947), 276-279. 
  5. ENRIQUES, F., Le superficie algebriche, Zanichelli, Bologna, 1949. 
  6. FULTON, W., Intersection Theory 2th Ed., Springer (1998). 
  7. GRIFFITHS, P. - HARRIS, J., Principles of Algebraic Geometry, Wiley and Sons, 1978. Zbl0408.14001
  8. MOISHEZON, B., Complex Surfaces and connected sums of complex projective planes, SpringerLect. Notes Math., 603 (1977), 1-234. Zbl0392.32015
  9. CILIBERTO, C. - MELLA, M. - RUSSO, F., Varierties with one apparent double point, to appear in Journal of Algebraic Geometry. Zbl1077.14076
  10. IONESCU, P., Embedded projective varieties of small invariants, SpingerL.N.M., 1056 (1984), 142-186. Zbl0542.14024
  11. IONESCU, P., Embedded projective varieties of small invariants. II, Rev. Roum. Math., 31 (1986), 539-545. Zbl0606.14038
  12. PIENE, R., A proof of Noether’s formula for the arithmetic genus of an algebraic surface, Compositio Math., 38 (1979), 113-119. Zbl0399.14004
  13. SEVERI, F., Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni e ai suoi punti tripli apparenti, Rend. Circ. Mat. Palermo, 15 (1901), 33-51. Zbl32.0648.04
  14. LAKSKOV, D., Residual intersections and Todd’s formula for the double locus of a morphism, Acta. Math., 140 (1978), 75-92. 
  15. LIVORNI, E. L., Classification of algebraic surfaces with sectional genus less than or equal to six. I: rational surfaces, Pacific J. of Math., 113 (1984), 93-114. Zbl0573.14013
  16. LIVORNI, E. L., Classification of algebraic surfaces with sectional genus less than or equal to six. II: Ruled surfaces with dim ϕ K X L ( X ) = 1 , Can. J. Math., Vol. XXXVIII, No. 5 (1986), 1110-1121. Zbl0598.14030
  17. LIVORNI, E. L., Classification of algebraic surfaces with sectional genus less than or equal to six. III: Ruled surfaces with dim φ K X L ( X ) = 1 , Math. Scand., 59 (1986), 9-29. Zbl0663.14024
  18. LIVORNI, E. L., Classification of algebraic non-ruled surfaces with sectional genus less or equal to six, Nagoya Math. J., Vol. 100 (1985), 1-9. Zbl0594.14028
  19. LIVORNI, E. L., On the existence of some surfaces, Lecture Notes in Math., 1417 (1990), Springer Verlag155-179. 
  20. EIN, L., Varieties with small dual variety I, Invent. Math., 86 (1989), 63-74. Zbl0603.14025

NotesEmbed ?

top

You must be logged in to post comments.