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Exercise

The geometric meaning of Corollary 18.17 in Eisenbud says is that given an equidimensional scheme $X$ of dimension $n$ and a surjective morphism $\pi:X\to V$ where $V$ is a smooth regular scheme of dimension $n$, then $X$ is locally CM Cohen Macaulay if and only if all fibers have the same length, or more precicely, that $\pi_* O_X$ is a free $O_V$ module. In fact, if $X$ is a projective CM variety of dimension $n$, there is always a finite map $\pi:X \to \mathbb{P}^n$ and any such map is flat. This means, for example, that $X$ cannot be extremely singular. A nice related result is that if $f:X\to Y$ is a morphism of projective varieties, with $X$ CM and $Y$ regular, and such that every fibre has dimension $\dim X-\dim Y$, then $f$ is flat. This is Exercise III.10.9 in Hartshorne.

As Sandor, Hailong and Karl points out below, normality implies CM in the case of surfaces. In general however, I don't think there are geometric properties that fully capture the Cohen-Macaulay property. I usually think of CM schemes as just schemes having similar properties as as locally complete intersections. For example, one has the following properties for a (Noetherian) Cohen-Macaulay scheme $X$:

i) If $X$ is generically reduced, then it is reduced. So in particular, $X$ has no embedded components.

ii) If $X$ is connected, it is connected in codimension 2, so that removing a closed subset of codimension $\ge 2$ is still connected.

iii) If $X$ has locally finite type, it is equidimensional.

(See Algebraic Geometry I, by Görtz , and Wedhorn). These properties are of course best for seeing which schemes are not CM, e.g., the standard non-example with the union of a plane and a line intersecting in a point is not CM, by iii).

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Exercise 18.17 in Eisenbud says that given an equidimensional scheme $X$ of dimension $n$ and a surjective morphism $\pi:X\to V$ where $V$ is a smooth scheme of dimension $n$, then $X$ is locally CM if and only if all fibers have the same length, or more precicely, that $\pi_* O_X$ is a free $O_V$ module. This means for example that $X$ cannot be extremely singular.

In general, I don't think there are geometric properties that fully capture the Cohen-Macaulay property. I usually think of CM schemes as schemes having similar properties as as locally complete intersections. For example, one has the following properties for a (Noetherian) Cohen-Macaulay scheme $X$:

i) If $X$ is generically reduced, then it is reduced. So in particular, $X$ has no embedded components.

ii) If $X$ is connected, it is connected in codimension 2, so that removing a closed subset of codimension $\ge 2$ is still connected.

iii) If $X$ has locally finite type, it is equidimensional.

(See Algebraic Geometry I, by Görtz, Wedhorn). These properties are of course best for seeing which schemes are not CM, e.g., the standard non-example with the union of a plane and a line intersecting in a point is not CM, by iii).

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Exercise 18.17 in Eisenbud says that given an equidimensional scheme $X$ of dimension $n$ and a surjective morphism $\pi:X\to V$ where $V$ is a smooth scheme of dimension $n$, then $X$ is CM if and only if all fibers have the same length, or more precicely, that $\pi_* O_X$ is a free $O_V$ module. This means for example that $X$ cannot be extremely singular.

In general, I don't think there are geometric properties that fully capture the Cohen-Macaulay property. I usually think of CM schemes as schemes having similar properties as as locally complete intersections. For example, one has the following properties for a (Noetherian) Cohen-Macaulay scheme $X$:

i) If $X$ is generically reduced, then it is reduced. So in particular, $X$ has no embedded components.

ii) If $X$ is connected, it is connected in codimension 2, so that removing a closed subset of codimension $\ge 2$ is still connected.

iii) If $X$ locally finite type, it is equidimensional.

(See Algebraic Geometry I, by Görtz, Wedhorn). These properties are of course best for seeing which schemes are not CM, e.g., the standard non-example with the union of a plane and a line intersecting in a point is not CM, by iii).