T-曲线的环境曲面与T-填充
Ambient Surfaces and T-Fillings of T-Curves
摘要：
T曲线是一种分段线性曲线，自20世纪90年代初以来，它已被成功地用于构造主要在实射影平面上具有预定拓扑的新的实代数曲线。事实上，T曲线可以用于任何真正的投影复曲面。我们在这里推广了后者的建设，离开非凸多边形，我们得到一类表面，周围的表面，我们的特征拓扑和T-曲线也很好地定义。与每一条T-曲线相关联，我们构造了一个曲面，即T-填充，它在拓扑上类似于实代数曲线通过复共轭的复化的商。然后利用T-填充证明了任意环境曲面上的T-曲线的一个定理，该定理类似于Harnack关于代数曲线的一个定理，并定义了T-曲线的可定向性和定向性，其定义方式与实代数中的定义方式相同。
英文摘要：
T-curves are piecewise linear curves which have been used with success since the beginning of the 1990's to construct new real algebraic curves with prescribed topology mainly on the real projective plane. In fact T-curves can be used on any real projective toric surface. We generalize here the construction of the latter by departing from non-convex polygons and we get a class of surfaces, the ambient surfaces, that we characterize topologically and on which T-curves are also well defined. Associated to each T-curve we construct a surface, the T-filling, which is topologically the analog of the quotient of the complexification of a real algebraic curve by the complex conjugation. We use then the T-filling to prove a theorem for T-curves on arbitrary ambient surfaces which is the analog of a theorem of Harnack for algebraic curves, and to define the orientability and orientation of a T-curve, the same way it is defined in the real algebraic setting.
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