# Tag Info

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### Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

RRT There is a big difference in difficulty between the compact Riemann surface case and the projective curve case, for reasons already mentioned. Namely a projective curve comes equipped with a ...
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### Is the divisibility graph of the proper divisors of n more often planar than not?

No, because almost all numbers have at least $4$ distinct prime factors, making the divisibility graph contain a hypercube and thus be nonplanar.
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### Are there topological versions of the idea of divisor?

Disclaimer. I am no expert at all in algebraic geometry. Therefore much of the following will be oversimplified or maybe even simply wrong. You are still invited to improve it. EDIT. There is a paper ...
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### How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function?

This is a well-known problem, attributed to Sam Wagstaff in Richard Guy's Unsolved Problems in Number Theory. Section B48 "Products taken over primes" includes a paragraph Wagstaff asked for an ...

### Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

Joe Harris, as recorded in his course notes here, gives the following slick proof when both $D$ and $K-D$ are effective; it has the advantage of never mentioning $H^1$. See lecture 1 for this argument,...

### On Q-Cartier Divisors

Maybe I can say something useful here. The main confusion seems to be how to find the sheaves/ideals/modules associated to multiples of divisors. As Martin Bright points out, symbolic power of a ...
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### Bertini's theorem over non-algebraically closed field

This is true both over finite and infinite fields. For infinite fields, see [Jou, Cor. I.6.11(2)]. It works for a general section of any very ample line bundle $\mathscr L$, using that over an ...
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### Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?

On page 73 of Kobayashi's book Hyperbolic Complex spaces he shows that if D is a certain configuration of 6 lines in the plane then its complement is complete hyperbolic and hyperbolically embedded ...
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### Generalization of the rigidity lemma in birational geometry

EDIT: I've just realized that this holds under somewhat weaker assumptions. It is not necessary that the fibers of $g$ are connected. EDIT#2: Apparently, in my previous edit I weakened the conditions ...
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### Fundamental groups of complements of divisors in $\mathbb P^2$

I'd leave this as a comment, but I don't have enough reputation. Consider the long exact sequence in homology of the pair $(\mathbb{P}^2, \mathbb{P}^2-D)$. Since $H_1(\mathbb{P}^2,\mathbb{Z}) = 0$ and ...
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### Anti-canonical divisor of a Fano variety

If you want to consider smooth (weak) Fano variety, then Fukuda has effective estimation of the birationality of anti-canonical systems for any dimension (but not optimal), see [S. FUKUDA, A note on ...
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### Is the reduced scheme associated to a Cartier divisor always Cartier?

No. Consider, for instance, the quadratic cone $$X = \{xz - y^2 = 0\} \subset \mathbb{A}^3$$ and the double line $$D = X \cap \{x = 0\} = \{x = y^2 = 0\}$$ on $X$. Then $D$ is a Cartier divisor, ...

### Is it possible to show that :for $n \geq 1:\sigma(n!-1)$ never be prime and why $\sigma(n!-1)\bmod 10$ at most is $0$?

If $n\ge 4$, then $24 \mid n!$. It is an easy exercise to show that if $24 \mid N$, then $24 \mid \sigma(N-1)$. (Pair each factor of $N-1$ with its cofactor, and use that every unit modulo $24$ is its ...
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### Push-forward of nef divisors via finite morphisms

I suppose you want $f$ to be surjective, otherwise $f_*D$ is not defined. Then $f_*D$ is nef: for any curve $C\subset Y$, $\ (f_*D\cdot C)=(D\cdot f^*C)\geq 0$. But it might very well be ample. ...
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