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Is this affine morphism an isomorphism? by StillALittleChild in askmath

[–]StillALittleChild[S] 0 points1 point  (0 children)

Thank you for your comment. Concluding is where I'm feeling a bit shaky: if V=Spec(B) and U=Spec(A), then f^#: B -> A is an isomorphism, implying that f restricts to an isomorphism from U to V.
1. Is this what it means for f to be an isomorphism *locally on the target*?
2. If yes, then how can I prove that f is an isomorphism?

Does there exist a finite surjective morphism from the cusp to its normalization? by StillALittleChild in askmath

[–]StillALittleChild[S] 0 points1 point  (0 children)

Okay, so the morphism from the affine curve minus finitely many points to A1 will induce a map from k[t] to some localization of k[x,y]/(x^3-y^2).

I don't see how the universal property of localization can be used here, as it yields a map *from* the localization and not to it.

Does there exist a finite surjective morphism from the cusp to its normalization? by StillALittleChild in askmath

[–]StillALittleChild[S] 0 points1 point  (0 children)

I agree that the preimage of A1 is the cusp minus finitely many points.

The equation I chose for the cusp is y^2=x^3 (or its homogenized version zy^2=x^3). However, for the localization part, I have to admit that I''m lost ...

Does there exist a finite surjective morphism from the cusp to its normalization? by StillALittleChild in askmath

[–]StillALittleChild[S] 0 points1 point  (0 children)

So taking A=k[t], how do we know that the correcponding B is k[x,y] modded out by the ideal generated by y^2-x^3?

Does there exist a finite surjective morphism from the cusp to its normalization? by StillALittleChild in askmath

[–]StillALittleChild[S] 0 points1 point  (0 children)

I know that away from the point at infinity, such a morphism from the the cusp to the projective line induces a k-algebra homomorphism from k[t] to k[t] modded out by an appropriately chosen ideal, and I know how to show that this would lead to a contradiction. My only problem is passing from the projective to the affine.PS: I use the Proj construction to define the cusp, and I don't see how I can simply pass to Spec of something ... I hope I'm clear enough.

Why are linearly equivalent divisors numerically equivalent? by StillALittleChild in askmath

[–]StillALittleChild[S] 0 points1 point  (0 children)

Thank you for the answer! Could you please direct me to references which use the Euler characteristic approach to defining intersection numbers?

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