Intereting Posts

roots of $f(z)=z^4+8z^3+3z^2+8z+3=0$ in the right half plane
Dirac delta of nonlinear multivariable arguments
Exponential growth of cow populations in Minecraft
Simultaneously Diagonalizable Proof
If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
Do real matrices always have real eigenvalues?
Möbius transformations on $D$ such that $f(D)=D$
How many non isomorphic groups of order 30 are there?
Completing an exact sequence
Simulate simple non-homogeneous Poisson proces
Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?
Example of an injective function $g$ and function $f$ such that $g\circ f$ is not injective
Prove inequality $\arccos \left( \frac{\sin 1-\sin x}{1-x} \right) \leq \sqrt{\frac{1+x+x^2}{3}}$
How to show that $\sin(n)$ does not converge?
Schmidt group and maximal subgroups

If $X$ is a complete nonsingular curve over $k$, $Y$ is any curve over $k$, $f: X \to Y$ is a morphism not map to a point (so $f(X)=Y$), then $f$ is a finite morphism.

This is the assertion prove in Hartshorne Chapter2, Prop6.8. But the proof is a little sketchy at the point of the inverse image of an affine set is also affine. I quote it here:

…Let $V=\rm{Spec}B$ be any open affine subset of $Y$. Let $A$ be the integral closure of $B$ in $K(X)$. Then $A$ is a finite $B$-module, and Spec$A$ is isomorphic to an open subset $U$ of $X$. Clearly $U=f^{-1}(V)$…

- Principal Bundles, Chern Classes, and Abelian Instantons
- Let K be a field, and $I=(XY,(X-Y)Z)⊆K$. Prove that $√I=(XY,XZ,YZ)$.
- irreducibility of a polynomial in several variables over ANY field
- Why degree of a reducible projective variety is the sum of the degree of its irreducible components
- Local ring at a non-singular point of a plane algebraic curve
- How to prove every radical ideal is a finite intersection of prime ideals?

Can anyone explain why “Spec$A$ is isomorphic to an open subset $U$ of $X$. Clearly $U=f^{-1}(V)$”?

- Why do we take the closure of the support?
- How many cubic curves are there?
- Can this quick way of showing that $K/(Y-X^2)\cong K$ be turned into a valid argument?
- The projection formula for quasicoherent sheaves.
- Homogeneous polynomial in $k$ can factor into linear polynomials?
- Intuitive definitions of the Orbit and the Stabilizer
- Proving $x^4+y^4=z^2$ has no integer solutions
- Surjective morphism of affine varieties and dimension
- Finite union of affine schemes
- Why do we need noetherianness (or something like it) for Serre's criterion for affineness?

Since $X$ is complete, the morphism $f$ is proper.

For any $y\in Y$, the fibre $F=f^{-1}(y) $ is closed and **strictly** included in $X$ , because $f$ is not constant.

Hence $F$ is finite, i.e. $f$ is quasi-finite.

But a proper and quasi-finite morphism is finite, and so $f$ is indeed a finite morphism.

- Infinite dimensional integral inequality
- No solutions to a matrix inequality?
- Find the limiting value of $S=a^{\sqrt{1}}+a^{\sqrt{2}}+a^{\sqrt{3}}+a^{\sqrt{4}}+…$ for $0 \leq a < 1$
- Can Euler's identity be extended to quaternions?
- Notation for modulo: congruence relation vs operator
- Proving that the estimate of a mean is a least squares estimator?
- Solving $f(x+y) = f(x)f(y)f(xy)$
- Summation of natural number set with power of $m$
- Proof of triangle inequality
- Prove if $f: \rightarrow \mathbb{R}$ is integrable, then so is $|f|$ and $\left|\int_{a}^{b}f(x)dx\right| \leq \int_{a}^{b}|f(x)|dx$
- Disjoint $AC$ equivalent to $AC$
- Largest modulus for Fermat-type polynomial
- Evaluating a logarithmic integral in terms of trilogarithms
- There is no isometry between a sphere and a plane.
- A non-increasing particular sequence