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A Sharp Castelnuovo Bound for the Normalization of Certain Projective Surfaces

In: Algebraic Geometry and its Applications

Author

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  • Nadia Chiarli

Abstract

Let P be the projective r-space (r ≥ 3) over an algebraically closed field k. Consider a reduced, irreducible, complete, non-degenerate surface X ⊆ P of degree d, and let $$\bar X$$ be its normalization: we shall assume $$\bar X$$ to be smooth (which is the case, e. g., when X is the generic projection of a given smooth projective surface $$\bar X$$ ). Let ∑(n) be the linear system of curves cut out on $$\bar X$$ by the hypersurfaces of P of degree n; in other words ∑(n) corresponds to the image of the canonical map $$\rho _n :H^0 (\text{P},O_\text{P} (n)) \to H^0 (\bar X,O_{\bar X} (nD))$$ (where D is the pull-back, via the normalization morphism $$\nu :\bar X \to X$$ , of the generic hyperplane section X′ of X).

Suggested Citation

  • Nadia Chiarli, 1994. "A Sharp Castelnuovo Bound for the Normalization of Certain Projective Surfaces," Springer Books, in: Chandrajit L. Bajaj (ed.), Algebraic Geometry and its Applications, chapter 7, pages 145-151, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2628-4_7
    DOI: 10.1007/978-1-4612-2628-4_7
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