Intereting Posts

An issue with approximations of a recurrence sequence
Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?
What is exactly meant by “preserves the group operation”?
Treating shocks with conservation laws
Non-standard models of arithmetic for Dummies
Why is $x^{1/n}$ continuous?
Why is every irreducible matrix with period 1 primitive?
Is it true that arcwise isometry that is also a local homeomorphism is a local isometry?
RSA solving for $p$ and $q$ knowing $\phi(pq)$ and $n$
Checking if a number is a Fibonacci or not?
Number-theoretic asymptotic looks false but is true?
How to prove Prime numbers can be expressed as $6k\pm 1$
How to evaluate integral using complex analysis
Expected Number of Coin Tosses to Get Five Consecutive Heads
Proof of injective and continuous

I am preparing for an exam in (mostly classical) algebraic geometry, and I have some preparatory questions, among which:

Can you write the equations of any nonsingular curve in any projective space which is not rational?

A problem with this question is that we never really defined what a “rational curve” is in class, but from what I can understand looking around, it should be a curve which is birationally equivalent to $\mathbb{CP}^1$.

- Affine Cover of $P^n$
- Is the restriction map of structure sheaf on an irreducible scheme injective?
- Find the latus rectum of the Parabola
- Why do we need noetherianness (or something like it) for Serre's criterion for affineness?
- When does variété mean manifold?
- Taking stalk of a product of sheaves

I have found this beautiful answer on MO, saying that cubic curves are an example since they have genus $1$ and $\mathbb{CP}^1\cong S^2$ has genus $0$. However, if I’m not mistaken, this relies on the fact that two smooth curves are birational iff they are isomorphic, which we didn’t see in class.

Is there some simple (and simple to prove) example for this question?

- Doing Complex Analysis on the Riemann Sphere?
- How to see $\operatorname{Spec} k$ for non necessarily algebraic closed field $k$?
- On limits, schemes and Spec functor
- An exercise in Liu regarding a sheaf of ideals (Chapter II 3.4)
- $t$ - th graded piece of the coordinate ring of $Y \times Z$
- Smoothness of $A \subseteq C$ implies smoothness of $B \subseteq C$? where $A\subseteq B \subseteq C$
- $V=\mathcal{Z}(xy-z) \cong \mathbb{A}^2$.
- Homogenous polynomials
- Does the fibres being equal dimensional imply flatness?
- Tensor product of reduced $k$-algebras must be reduced?

The Fermat curve $X:=V(f) \subset \mathbb P^2$ with equation $$f(x,y,z) = x^d + y^d + z^d, d \in \mathbb N,$$ is smooth with geometric genus $g(X) = \frac {(d-1)*(d-2)}{2}$. The curve X is not rational iff $g(X) > 0$, i.e. iff degree $d > 2$.

A curve is *rational* by definition iff it is birational equivalent to projective space $\mathbb P^1$. **Note**. A smooth projective curve is rational iff it is biholomorphic to $\mathbb P^1$.

- Prove $\sqrt{2} + \sqrt{5}$ is irrational
- Norm of the Resolvent
- Is each power of a prime ideal a primary ideal?
- question about typical proof of Krull Intersection Theorem
- Variation of the Kempner series – convergence of series $\sum\frac{1}{n}$ where $9$ is not a digit of $1/n$.
- Proof of Inequality using AM-GM
- Solving for $f$
- Complex number trigonometry problem
- Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares
- Two questions about equivalence relations
- For what values of $x$ is $\cos x$ transcendental?
- How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$?
- Give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain
- What is importance of the Bunyakovsky conjecture?
- Sentences in first order logic for graphs