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.

我们称平滑不可约投影曲线为 Castelnuovo 曲线，如果它允许双有理映射进入投影 r 空间，使得图像曲线具有至少 2r+1 的度数和最大可能的几何亏格（可以通过经典公式计算，由于到新堡）。众所周知，Castelnuovo 曲线必须位于 Hirzebruch 曲面（有理直纹曲面）上。相反，利用 W. Castryck 和 F. Cools 关于 Hirzebruch 曲面上曲线的滚动不变量的结果，我们表明 Hirzebruch 曲面上的曲线是 Castelnuovo 曲线，除非它们的亏格因它们的角性而变得太小。我们更仔细地分析情况，并根据 g 和 k 计算位于 Hirzebruch 曲面上的固定亏格 g 和固定角性 k 的曲线的模数。