2

I was reading Judy Walker's book Codes and Curves, and one of the exercise's in the book (ex. 4.6) was proving that every non-singular curve is abaolutely irreducible.

I'm not so familiar with algebric geometry, and I'm not so sure how to approach this (beside playing a bit with the definitions, which reached a dead end).

Thank you!

Yotam D
  • 369

1 Answers1

5

If a curve $\cal C$ is not absolutely irreducible, it will split as ${\cal C}={\cal C}_1\cup{\cal C}_2$ over some algebraically closed field. Then every point in the non-empty intersection ${\cal C}_1\cap{\cal C}_2$ would be a singular (non-smooth) point of ${\cal C}$

AdLibitum
  • 3,003
  • Thank you for the answer. Why is the intersection non-empty? – Yotam D Aug 15 '15 at 08:45
  • Over an algebraically closed field two projective palne curves always intersect. Check the theory of resultants (https://en.wikipedia.org/wiki/Resultant) or, if you want a big shot, Bezout's theorem (https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem) – AdLibitum Aug 15 '15 at 08:50