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Ben Webster ♦
Ben Webster ♦
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A very well-known condition is that the Hilbert scheme of a smooth surface is smooth (I believe also. As David pointed out below, the Hilbert scheme of a smooth curve is smooth and equal to the symmetric product (since k[t] has only one finite dimension quotient of each dimension).

I don't know of any other examples, but one of the versions of Murphy's Law in algebraic geometry is roughly "if you don't have a good reason for a Hilbert scheme to not be horrible, it will be as horrible as you can possibly imagine."

A very well-known condition is that the Hilbert scheme of a smooth surface is smooth (I believe also the Hilbert scheme of a smooth curve is smooth).

I don't know of any other examples, but one of the versions of Murphy's Law in algebraic geometry is roughly "if you don't have a good reason for a Hilbert scheme to not be horrible, it will be as horrible as you can possibly imagine."

A very well-known condition is that the Hilbert scheme of a smooth surface is smooth. As David pointed out below, the Hilbert scheme of a smooth curve is smooth and equal to the symmetric product (since k[t] has only one finite dimension quotient of each dimension).

I don't know of any other examples, but one of the versions of Murphy's Law in algebraic geometry is roughly "if you don't have a good reason for a Hilbert scheme to not be horrible, it will be as horrible as you can possibly imagine."

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Ben Webster ♦
Ben Webster ♦
  • 44.7k
  • 12
  • 126
  • 260

A very well-known condition is that the Hilbert scheme of a smooth surface is smooth (I believe also the Hilbert scheme of a smooth curve is smooth).

I don't know of any other examples, but one of the versions of Murphy's Law in algebraic geometry is roughly "if you don't have a good reason for a Hilbert scheme to not be horrible, it will be as horrible as you can possibly imagine."