On curves on Hirzebruch surfaces

IF 0.4 4区 数学 Q4 MATHEMATICS
Gerriet Martens
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引用次数: 0

Abstract

We call a smooth irreducible projective curve a Castelnuovo curve if it admits a birational map into the projective r-space such that the image curve has degree at least 2r+1 and the maximum possible geometric genus (which one can calculate by a classical formula due to Castelnuovo). It is well known that a Castelnuovo curve must lie on a Hirzebruch surface (rational ruled surface). Conversely, making use of a result of W. Castryck and F. Cools concerning the scrollar invariants of curves on Hirzebruch surfaces we show that curves on Hirzebruch surfaces are Castelnuovo curves unless their genus becomes too small w.r.t. their gonality. We analyze the situation more closely, and we calculate the number of moduli of curves of fixed genus g and fixed gonality k lying on Hirzebruch surfaces, in terms of g and k.

在Hirzebruch曲面的曲线上
我们称光滑不可约投影曲线为Castelnuovo曲线,如果它在投影r空间中允许一个双象映射,使得图像曲线具有至少2r+1的度和最大可能的几何属(可以通过Castelnuovo的经典公式计算)。众所周知,Castelnuovo曲线必须位于Hirzebruch曲面(有理直纹曲面)上。相反地,利用W. Castryck和F. cool关于Hirzebruch曲面上曲线的涡旋不变量的结果,我们证明了Hirzebruch曲面上的曲线是Castelnuovo曲线,除非它们的属相对于它们的共向性变得太小。我们更仔细地分析了这种情况,并计算了Hirzebruch曲面上固定格g和固定向性k曲线的模数,用g和k表示。
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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.
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