AI 中文总结
该研究提出新构造方法，加深有理曲线与反典范除子正性的联系，证明具大藤田不变量的簇有大型有理曲线族，构造奇异法诺簇上的自由有理曲线，还推导出法诺超曲面上曲线空间维数的结论。
AI 中文摘要
我们提出一种新的构造方法，可从高亏格曲线族中拆分出大度的有理曲线。该构造与结果加深了有理曲线与反典范除子正性之间的联系。具体而言，我们证明具有大藤田不变量的簇容许有理曲线的大型族，还在某些奇异法诺簇上构造了自由有理曲线。作为结果的明确推论，我们证明对于指数至少为3的一般法诺超曲面，所有亏格g且度数足够大的曲线空间均具有预期维数。
英文摘要
We present a new construction that allows us to break off large-degree rational curves from families of higher genus curves. Our construction and results deepen the connection between rational curves and positivity of the anticanonical divisor. Specifically, we show that varieties with large Fujita invariant admit large families of rational curves. We also construct free rational curves on certain singular Fano varieties. As an explicit consequence of our results, we prove that for a general Fano hypersurface of index at least 3, all spaces of genus g curves of sufficiently large degree have the expected dimension.
Comments48 pages
