论文标题
希尔伯特（Hilbert
Components of the Hilbert Scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
For $γ\geq 7$ and $g \geq 6γ+ 5$, we construct a family $\mathcal{F}^{\prime}$ of curves lying on cones in $\mathbb{P}^{g-3γ+1}$ over smooth non-degenerate curves of genus $γ$ and degree $g-2γ$ in $\mathbb{P}^{g-3γ+1}$. We show that $\dim \mathcal{F}^{\prime} = 2g-γ-1 + (g-3γ+1)^2$. For a general curve $X^{\prime}$ from the family $\mathcal{F}^{\prime}$, we compute the dimension of the space of its first-order deformations. We prove that the family $\mathcal{F}^{\prime}$ gives rise to an irreducible, non-reduced component $\mathcal{D}^{\prime}$ of the Hilbert scheme $\mathcal{I}_{2g-4γ+ 1, g, g - 3γ+ 1}$, which parametrizes smooth, irreducible, non-degenerate curves of degree $2g-4γ+ 1$ and genus $g$ in $\mathbb{P}^{g-3γ+1}$. We obtain $\dim T_{[X^{\prime}]} \mathcal{D}^{\prime} = \dim \mathcal{D}^{\prime} + 1 = \dim \mathcal{F}^{\prime} + 1$.
