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Lester
Lester
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This is a very basic question, but I can't find a clean answer anywhere.

In introductory algebraic geometry books working over the complex numbers, it's usual to use these three words interchangeably. A point on a variety $X$ is smooth/regular/nonsingular if the dimension of the tangent space at the point is equal to the dimension of the variety.

On the other hand, I know that people sometimes find it important to distinguish between these terms, maybe when defining smoothness of morphisms, or working over non-closed fields,...

I want to make sure I know the right definitions of these terms in current use. In what contexts should each be defined? What implies what? How should I think of them?

Edit: See for example https://en.wikipedia.org/wiki/Regular_scheme , which says there are regular schemes that aren't smooth. There are also these notes of Vakil, where he has crossed out "smooth" and replaced it with "nonsingular": https://math.stanford.edu/~vakil/0708-216/216class21.pdf The notes seem to suggest it's because "smooth" is reserved as a property of morphisms. Is there a reason Wiki is happy to say "smooth scheme" but Ravi isn't? Is "nonsingular" the same as "regular"?

This is a very basic question, but I can't find a clean answer anywhere.

In introductory algebraic geometry books working over the complex numbers, it's usual to use these three words interchangeably. A point on a variety $X$ is smooth/regular/nonsingular if the dimension of the tangent space at the point is equal to the dimension of the variety.

On the other hand, I know that people sometimes find it important to distinguish between these terms, maybe when defining smoothness of morphisms, or working over non-closed fields,...

I want to make sure I know the right definitions of these terms in current use. In what contexts should each be defined? What implies what? How should I think of them?

This is a very basic question, but I can't find a clean answer anywhere.

In introductory algebraic geometry books working over the complex numbers, it's usual to use these three words interchangeably. A point on a variety $X$ is smooth/regular/nonsingular if the dimension of the tangent space at the point is equal to the dimension of the variety.

On the other hand, I know that people sometimes find it important to distinguish between these terms, maybe when defining smoothness of morphisms, or working over non-closed fields,...

I want to make sure I know the right definitions of these terms in current use. In what contexts should each be defined? What implies what? How should I think of them?

Edit: See for example https://en.wikipedia.org/wiki/Regular_scheme , which says there are regular schemes that aren't smooth. There are also these notes of Vakil, where he has crossed out "smooth" and replaced it with "nonsingular": https://math.stanford.edu/~vakil/0708-216/216class21.pdf The notes seem to suggest it's because "smooth" is reserved as a property of morphisms. Is there a reason Wiki is happy to say "smooth scheme" but Ravi isn't? Is "nonsingular" the same as "regular"?

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Lester
Lester
  • 371
  • 1
  • 4
  • 5

Smooth vs regular vs non-singular

This is a very basic question, but I can't find a clean answer anywhere.

In introductory algebraic geometry books working over the complex numbers, it's usual to use these three words interchangeably. A point on a variety $X$ is smooth/regular/nonsingular if the dimension of the tangent space at the point is equal to the dimension of the variety.

On the other hand, I know that people sometimes find it important to distinguish between these terms, maybe when defining smoothness of morphisms, or working over non-closed fields,...

I want to make sure I know the right definitions of these terms in current use. In what contexts should each be defined? What implies what? How should I think of them?