论文标题
平面代数曲线具有规定的奇异性
Plane algebraic curves with prescribed singularities
论文作者
论文摘要
我们报告了平面中存在复杂和真实代数曲线的问题，其规定的奇异性涉及分析和拓扑等效性。问题是，对于给定的积极整数$ d $以及有限数量的给定分析或拓扑奇异性类型，是否存在具有$ d $的平面曲线（不可简化的）曲线，具有给定类型的奇异点是其唯一的奇异性。所有此类曲线的集合都是准标准品种，我们称之为时尚家族（ESF）。我们用曲线的数值不变性及其奇异性来描述，与相应ESF的非空性和$ t $平滑度（即预期维度平滑）有关的必要条件和足够的条件。所考虑的奇异性可以是任意的，但是我们将节点和尖齿的平面曲线特别关注，这是研究最多的案例，总体上仍然没有完整的答案。但是，重要的结果是，如果该学位获得无穷大，则必要和足够的条件对$ t $ smoth equinardular的家庭显示出相同的渐近学。
We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree $d$ having singular points of the given type as its only singularities. The set of all such curves is a quasi-projective variety, which we call an equisingular family (ESF). We describe, in terms of numerical invariants of the curves and their singularities, the state of the art concerning necessary and sufficient conditions for the non-emptiness and $T$-smoothness (i.e., smooth of expected dimension) of the corresponding ESF. The considered singularities can be arbitrary, but we spend special attention to plane curves with nodes and cusps, the most studied case, where still no complete answer is known in general. An important result is, however, that the necessary and the sufficient conditions show the same asymptotics for $T$-smooth equisingular families if the degree goes to infinity.
