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Questions tagged [divisors-algebraic-geometry]

For questions involving Cartier and Weil divisors, the Riemann-Roch theorem and related topics (e.g. Chern classes and line bundles) on algebraic varieties.

For questions involving divisors on algebraic varieties.

The Riemann-Roch theorem states

$$\ell(D)-\ell(K-D) = \deg(D) - g + 1$$

where $D$ is a divisor, $K$ is a canonical divisor and $g$ is the genus of the surface.

543 questions
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Linear System Attached To Projective Embedding

I've been doing some reading on divisors (Chapter 4 of Murty's book on Abelian Varieties) and I have a fairly elementary question I've spent far too much time trying to figure out. Many authors treat the equivalence between (certain) linear systems…
Rdrr
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Find the divisors of $2$ in $R$.

Task: In the ring $R: = Z [i]$ we have to show: Find the divisors of $2$ in $R$. (Hint: Use the norm $N (α) = a^2 + b^2$ for $α = a + bi ∈ R$.) Attempt: So I can already determine the number of divisors: In $Z [i]$, $2$ has the decomposition $2 = (1…
Vek
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Explicit description of sheaf of differentials on $\mathbb{P}^1_k$ using affine charts, and corresponding Cartier Divisor

Let $k$ be a field. We define the scheme $X = \mathbb{P}^1_k$ to be the gluing of the affine schemes $\text{spec}(k[T])$ and $\text{spec}(k[U])$ via the isomorphism \begin{align*} \phi: k[T,T^{-1}] &\to k[U,U^{-1}] \\ T &\mapsto…
Improve
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Efficently find divisors of numbers up to 20 digits

I am aware that finding divisors of large numbers is a well known mathematical problem (which is one of the reasons why cryptography works). Most solutions I've stumbled across used prime factorization to speed up the process a bit. However, on…
Kyu96
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