论文标题
具有许多规定的组件的真实代数超曲面的存在
Existence of real algebraic hypersurfaces with many prescribed components
论文作者
论文摘要
给定一个真正的代数$ x $ dimension $ n $，$ x $上的一个非常丰富的除数$ d $，以及一个流畅的封闭的超浮力$σ$ $ \ mathbf {r}^n $，我们构建了真实的代数超级表面，在线性的linear locus $ | MD | MD | MD | $ compus compus $ | MD | $ compus $ | $ compus $ | $ compus $ compus $ compus $ compus $ compus $ n $。结果，我们表明了线性系统中的真实代数超曲面的存在$ | md | $，其betti数字按最大订单增长，因为$ m $ to to in infinity。作为另一个应用程序，我们在$ \ mathbf {c} \ Mathbf {p}^n $的任何平滑复杂的超脸中恢复了许多不相交拉格朗日的结果。本文中的结果更为普遍地用于完整的交叉点。我们主要结果的证明使用了概率工具。
Given a real algebraic variety $X$ of dimension $n$, a very ample divisor $D$ on $X$ and a smooth closed hypersurface $Σ$ of $\mathbf{R}^n$, we construct real algebraic hypersurfaces in the linear system $|mD|$ whose real locus contains many connected components diffeomorphic to $Σ$. As a consequence, we show the existence of real algebraic hypersurfaces in the linear system $|mD|$ whose Betti numbers grow by the maximal order, as $m$ goes to infinity. As another application, we recover a result by D. Gayet on the existence of many disjoint lagrangians with prescribed topology in any smooth complex hypersurface of $\mathbf{C}\mathbf{P}^n$. The results in the paper are proved more generally for complete intersections. The proof of our main result uses probabilistic tools.
