Intereting Posts

Do the spaces spanned by the columns of the given matrices coincide?
negative multiple of ample line bundle has no global section
Formula for prime counting function
Yitang Zhang: Prime Gaps
Multivariable Epsilon-Delta Proof?
Where is the form $f(g(x))g'(x)$?
Elementary solution of exponential Diophantine equation $2^x – 3^y = 7$.
Show that $C_1= [\frac{k}{2^n},\frac{k+1}{2^n})$ generates the Borel σ-algebra on R.
How to prove that $(A\cup B) \cap C \subseteq A\cup (B\cap C)$
Combinatorial interpretation for the identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$?
Cutting a cube by plane cuts
arrangement of NOT sitting together
Inequality Of Four Variables
Ring without distributive law?
Function which is not in $L^2(R^n)$

For an affine variety $X\subset \mathbb{A}^n$, we can associate it with $\overline{X}$, which is the closure of $X$ in $\mathbb{P}^n$.

Does $\overline{X}$ depend on the choice of embedding $X\subset\mathbb{A}^n

$? Suppose given $f\colon X\subset\mathbb{A}^n\to Y\subset\mathbb{A}^n$，does it induce morphism $\overline{f}\colon \overline{X}\to \overline{Y}$?

- Quick question: Direct sum of zero-dimensional subschemes supported at the same point
- Exercise $1.8$ of chapter one in Hartshorne.
- Isomorphism of Proj schemes of graded rings, Hartshorne 2.14
- Which functor does the projective space represent?
- Cardinality and degrees of irreducible components of an affine variety
- Applications of Belyi's theorem

- Local ring at a non-singular point of a plane algebraic curve
- Going down theorem fails
- What is the ring of global sections of this variety?
- When a holomorphy ring is a PID?
- What is the differences between scheme-theoretical intersection and set-theoretical intersection?
- Tangent space of a point of an algebraic variety
- Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.
- Picard group and cohomology
- Is there an equation that will graph a line segment?
- Example I.4.9.1 in Hartshorne (blowing-up)

Yes, the projective closure depends on the choice of embedding. For example, the closure of $\{ x^3 – y = 0 \} \subset \mathbb{A}^2$ in $\mathbb{P}^2$ is singular, but it can also be embedded in a non-singular projective curve.

Functoriality also fails in general: the morphism $\{ x^3 – y = 0 \} \subset \mathbb{A}^2 \to \mathbb{A}^1$ given by $(x, y) \mapsto x$ cannot be extended to a morphism between the projective closures.

- How Differential got into calculus
- Are weak derivatives and distributional derivatives different?
- I understand what a division ring is, but cannot find any examples. Can anyone give me one?
- If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?
- Prove the product of a polynomial function of the roots of another polynomial is an integer.
- prove that : cosx.cos(x-60).cos(x+60)= (1/4)cos3x
- On the limits of weakly convergent subsequences
- If there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$
- Why can't you add apples and oranges, but you can multiply and divide them?
- Find the sum of digits in decimal form of the given number
- Is $\lim_{n\to \infty} \frac{np_n}{\sum_{i=1}^n p_i} = 2$ true?
- Reduced cost in the Phase II of the two-phase Simplex?
- The Last Man Standing
- Proof/Intuition for Eigenvalues to Solve Linear Differential Equations
- Count subsets of an i-set in two ways