A projective hypersurface $X \subseteq \mathbb P^n$ has defect if $h^i(X) \neq h^i(\mathbb P^n)$ for some $i \in \{n, \dots, 2n-2\}$ in a suitable cohomology theory. This occurs for example when $X \subseteq \mathbb P^4$ is not $\mathbb Q$-factorial. We show that in characteristic 0, the Tjurina number of hypersurfaces with defect is large. For $X$ with mild singularities, there is a similar result in positive characteristic. As an application, we obtain a lower bound on the asymptotic density of hypersurfaces without defect over a finite field.

中文翻译：

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有缺陷的超曲面

一个射影超曲面 $X \subseteq \mathbb P^n$ 有缺陷如果 $h^i(X) \neq h^i(\mathbb P^n)$ 对于某些 $i \in \{n, \dots, 2n -2\}$ 在合适的上同调理论中。例如，当 $X \subseteq \mathbb P^4$ 不是 $\mathbb Q$-factorial 时，就会发生这种情况。我们表明，在特征 0 中，具有缺陷的超曲面的 Tjurina 数很大。对于具有轻度奇异性的 $X$，正特性也有类似的结果。作为一个应用，我们获得了在有限域上无缺陷的超曲面渐近密度的下界。