Remark 111.50.2. The $k$th Fitting ideal of $\Omega _{X/S}$ is commonly used to define the singular scheme of the morphism $X \to S$ when $X$ has relative dimension $k$ over $S$. But as part (a) shows, you have to be careful doing this when your family does not have “constant” fibre dimension, e.g., when it is not flat. As part (b) shows, flatness doesn't guarantee it works either (and yes this is a flat family). In “good cases” – such as in (c) – for families of curves you expect the $0$-th Fitting ideal to be zero and the $1$st Fitting ideal to define (scheme-theoretically) the singular locus.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)