2 added 318 characters in body

What is the geometric meaning of Cohen-Macaulay schemes?

Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various nice operations. But whereas complete intersections have an obvious geometric meaning, I don't know if this is true for CM schemes. Perhaps we can make somethinkg out of the following theorem: A noetherian ring $R$ is CM iff every ideal $I$ which can be generated by $ht(I)$ many elements is unmixed, i.e. has no embedded associated primes. Also, Eisenbud suggests that Cor. 18.17 in his book "Commutative algebra with a view toward algebraic geometry" reveals some kind of geometric meaning, but perhaps someone can explain this in detail?

Every integral curve is CM. Now assume that you are given a surface, given by some homogeneous equations, how do you "see" if it is CM or not?

Hailong's answer contains the link to http://mathoverflow.net/questions/6704, which is pretty the same question as mine and has already some very good answers. So I appologize for this duplicate. But the answers here reveal some more insights, thanks!

1

# Geometric meaning of Cohen-Macaulay schemes

What is the geometric meaning of Cohen-Macaulay schemes?

Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various nice operations. But whereas complete intersections have an obvious geometric meaning, I don't know if this is true for CM schemes. Perhaps we can make somethinkg out of the following theorem: A noetherian ring $R$ is CM iff every ideal $I$ which can be generated by $ht(I)$ many elements is unmixed, i.e. has no embedded associated primes. Also, Eisenbud suggests that Cor. 18.17 in his book "Commutative algebra with a view toward algebraic geometry" reveals some kind of geometric meaning, but perhaps someone can explain this in detail?

Every integral curve is CM. Now assume that you are given a surface, given by some homogeneous equations, how do you "see" if it is CM or not?