rational normal scroll
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Author(s):  
Vincenzo Di Gennaro

AbstractLet $$(S,{\mathcal {L}})$$ ( S , L ) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $${\mathcal {L}}$$ L of degree $$d > 25$$ d > 25 . In this paper we prove that $$\chi (\mathcal O_S)\ge -\frac{1}{8}d(d-6)$$ χ ( O S ) ≥ - 1 8 d ( d - 6 ) . The bound is sharp, and $$\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$$ χ ( O S ) = - 1 8 d ( d - 6 ) if and only if d is even, the linear system $$|H^0(S,{\mathcal {L}})|$$ | H 0 ( S , L ) | embeds S in a smooth rational normal scroll $$T\subset {\mathbb {P}}^5$$ T ⊂ P 5 of dimension 3, and here, as a divisor, S is linearly equivalent to $$\frac{d}{2}Q$$ d 2 Q , where Q is a quadric on T. Moreover, this is equivalent to the fact that a general hyperplane section $$H\in |H^0(S,{\mathcal {L}})|$$ H ∈ | H 0 ( S , L ) | of S is the projection of a curve C contained in the Veronese surface $$V\subseteq {\mathbb {P}}^5$$ V ⊆ P 5 , from a point $$x\in V\backslash C$$ x ∈ V \ C .


2016 ◽  
Vol 27 (05) ◽  
pp. 1650045 ◽  
Author(s):  
Danielle Lara ◽  
Simone Marchesi ◽  
Renato Vidal Martins

Let [Formula: see text] be an integral and projective curve whose canonical model [Formula: see text] lies on a rational normal scroll [Formula: see text] of dimension [Formula: see text]. We mainly study some properties on [Formula: see text], such as gonality and the kind of singularities, in the case where [Formula: see text] and [Formula: see text] is non-Gorenstein, and in the case where [Formula: see text], the scroll [Formula: see text] is smooth, and [Formula: see text] is a local complete intersection inside [Formula: see text]. We also prove that the canonical model of a rational monomial curve with just one singular point lies on a surface scroll if and only if the gonality of the curve is at most [Formula: see text], and that it lies on a threefold scroll if and only if the gonality is at most [Formula: see text].


2011 ◽  
Vol 18 (01) ◽  
pp. 121-128
Author(s):  
Margherita Barile

We show that a rational normal scroll can in general be set-theoretically defined by a proper subset of the 2-minors of the associated two-row matrix. This allows us to find a class of rational normal scrolls that are almost set-theoretic complete intersections.


1999 ◽  
Vol 22 (3) ◽  
pp. 489-496
Author(s):  
Cícero F. Carvalho

We study the distribution of the total and ordinary ramification points of a trigonal curve over the intersection of this curve with rational curves on a rational normal scroll. We show, among other results, that these intersections may contain all the ramification points of the trigonal curve.


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