Linear pencils encoded in the Newton polygon
Linear pencils encoded in the Newton polygon
复制标题
以牛顿多边形编码的线性铅笔
DOI：
10.1093/imrn/rnw082
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Filip Cools
中科院分区：
文献类型：
--
作者：
W. Castryck;Filip Cools
设$C$是一个代数曲线定义的一个充分一般的二元洛朗多项式与给定的牛顿多边形$\Delta$。经典的是$C$的几何亏格等于$\Delta$内部的格点数。本文通过去掉川口最近的一个结果中的一个技术性假设，给出了一致性、Clifford指数和Clifford维数的类似的组合解释.更一般地说，该方法表明，除了某些众所周知的例外，每一个基点自由铅笔的程度等于或略超过gonality是“组合”，在这个意义上，它对应于投影$C$沿着一个晶格方向。然后，我们给出了一个解释与组合铅笔的滚动不变量，并显示如何可以告诉铅笔是否是完整的。在这些应用中，我们发现每一条光滑射影曲线至多有一个嵌入维数为2的Weierstrass半群，并且如果亏格为2的非超椭圆光滑射影曲线C可以嵌入到第n个Hirzebruch曲面H n中，则H n实际上是C的不变量.
Let $C$ be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon $\Delta$. It is classical that the geometric genus of $C$ equals the number of lattice points in the interior of $\Delta$. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is 'combinatorial', in the sense that it corresponds to projecting $C$ along a lattice direction. We then give an interpretation for the scrollar invariants associated to a combinatorial pencil, and show how one can tell whether the pencil is complete or not. Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension $2$, and that if a non-hyperelliptic smooth projective curve $C$ of genus $g \geq 2$ can be embedded in the $n$th Hirzebruch surface $\mathcal{H}_n$, then $n$ is actually an invariant of $C$.
影响因子：
0.9
作者：
P. Wilson
通讯作者：
P. Wilson