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

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Let X ⊂ ℙ 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 ℙ 4 , in ℙ 5 and a conic bundle of degree 5 in ℙ 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

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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

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  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
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