Intereting Posts

Mathematical trivia (i.e. collections of anecdotes and miscellaneous (recreational) mathematics)
proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
Big list of serious but fun “unusual” books
The inner product determines the structure of the space
If $f$ is continuous in $$, find $\lim\limits_{x \to 0^{+}}x\int\limits_x^1 \frac{f(t)}t dt$
Conditional mean on uncorrelated stochastic variable
Proving limit with $\log(n!)$
How can we show that 3-dimensional matching $\le_p$ exact cover?
Given every horse's chance of winning a race, what is the probability that a specific horse will finish in nth place?
A question regarding Markov Chains
Most ambiguous and inconsistent phrases and notations in maths
Component of a vector perpendicular to another vector.
Reconciling two different definitions of constructible sets
The alternating group is a normal subgroup of the symmetric group
Number of solutions to a set of homogeneous equations modulo $p^k$

I have some difficulties to prove that the image of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that

$$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$

is the algebraic projective set

$$V(XT-YZ,\, Y^2-XZ,\,Z^2-YT)$$

Clearly I have problem to prove that the algebraic projective set is contained in the image of $f$. In particular “solving brutally” the polynomial system I’m losing my mind in calculations, so I hope that there is a simpler method.

- Proof of $X(\mathcal{O}_K)\simeq X_K(K)$
- The Picard Group of the Affine line with double origin
- Intersection of open affines can be covered by open sets distinguished in *both*affines
- negative multiple of ample line bundle has no global section
- Why is the Artin-Rees lemma used here?
- Determining the generators of $I(X)$
- On a certain morphism of schemes from affine space to projective space.
- There is a bijection between irreducible components of the generic fiber and irreducible components passing through it.
- Extended ideals and algebraic sets
- When can stalks be glued to recover a sheaf?

Every point of $V=V(XT-YZ,\, Y^2-XZ,\,Z^2-YT)$ is in at least one of the four standard copies of $\mathbb A^3$affine spaces covering $\mathbb P^3$. So check them successively. I’ll do the $T=1$ part.

When $T=1$, $V$ is given by $$x=yz, \: y^2=xz,\: z^2=y$$

But then a point $(x,y,z)=[x:y:z:1] \in V $ satisfying these equations is simply the image under $f$ of $[u:v]=[z:1]\in \mathbb P^1 $, since the first and third displayed equations for $V$ immediately imply that $x=yz=z^2.z=z^3$

- What is a simple example of a limit in the real world?
- Euclidean Geometry Intersection of Circles
- Quadratic Variation of Brownian motion indexed by cadlag increasing function
- Are “most” continuous functions also differentiable?
- A question about Darboux functions
- Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}$?
- Strategies to denest nested radicals.
- Sequence Of Primes
- How to prove $\sum\limits_{i=0}^{\lfloor\frac{r}{2}\rfloor}\binom{r}{i}\binom{r-i}{r-2i}2^{r-2i}=\binom{2r}{r}$
- Proof that the intersection of any finite number of convex sets is a convex set
- Set theory practice problems?
- RSA: Creating a key of desired length
- Finding polynomial given the remainders
- On the closed form for $\sum_{m=0}^\infty \prod_{n=1}^m\frac{n}{5n-1}$
- solution set for congruence $x^2 \equiv 1 \mod m$