Spaces to Rings and Back Again!
- Given any space X, the set R(X, k) of maps from X to a field k
is itself a ring.
- Given a ring A, the set Spec(A) of prime ideals of A is a space.
- In fact, if you set things up right, R(Spec(A), k) = A and
Spec(R(X, k)) = X.
In Algebraic Geometry, we study spaces by their rings and rings by their spaces!
- The equation y2 = x3 + x2 - 2x defines an elliptic curve in the real plane.
- You get more solutions over the complex numbers.
- Here's what the rest of it looks
like over the complex numbers.
- The ring you get from the elliptic curve is C[x,y]/(y2 - x3 - x2 + 2x); the space you get from C[x,y]/(y2 - x3 - x2 + 2x) is the elliptic curve!
For more, see:
- B. Harbourne, Seshadri constants and very ample divisors on algebraic surfaces, J. Reine Angew. Math. 559 (2003) 115--122.
- A. Seceleanu, Inverse systems, fat points and the weak Lefschetz property, joint with B. Harbourne and H. Schenck, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 712-730.
J. Walker, Codes and Curves, Amer. Math. Soc. book, 2000
- S. Wiegand, Projective lines over one-dimensional semilocal domains and spectra of birational extensions, in: Algebra Geometry and Applications, (C. Bajaj, ed.), Springer Verlag, New York 1994.
- W. Zhang, Lyubeznik numbers of projective schemes, Adv. Math. 228 (2011), 575--616.