8
votes
2answers
282 views
Proving that a generic variety with ample canonical bundle has no automorphisms
Let $X$ be a smooth projective connected variety over the complex numbers with ample canonical bundle. If $X$ is generic and $\dim X \leq1$, the automorphism group of $X$ is trivia …
8
votes
0answers
106 views
Infinitely many curves with isogenous Jacobians
Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?
Does the situation change in positive characteristic?
1
vote
0answers
131 views
Complete curves in $M_g$ and Theta Characteristics
Let $g\geq 3$. Following the reference below, the locus of curves in $M_g$ with an effective even theta characteristic has codimension $1$. (Those are the curves $C$ with an effect …
6
votes
0answers
66 views
Permutations of prescribed cycle types that multiply to the identity
Suppose that $\lambda_1,\lambda_2,\lambda_3$ are partititions of $n$. When do there exist permutations $\sigma_1,\sigma_2,\sigma_3 \in S_n$ such that
(1) $\sigma_1\sigma_2\sigma_ …
16
votes
0answers
238 views
Tame morphism from a curve to $\mathbb{P}^1$
Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $C$ be a smooth projective curve over $k$. Is it possible to find a map $C \to \mathbb{P}^1$ that is tamely …
0
votes
0answers
107 views
canonical model of a reducible curve
Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the ( …
3
votes
0answers
79 views
state of the art for kodaira dimension of $\overline{\mathcal{M}}_{g,n}$
What is the state of the art about the kodaira dimension (and rationality, unirationality, etc.) of the moduli spaces of $n$-pointed curves of genus $g$? When it is known and when …
1
vote
1answer
138 views
Are the projection morphisms from a product of varieties necessarily open?
If C and D are irreducible, affine varieties over an algebraically closed field, and I form the product variety CxD, is the projection morphism from CxD to C necessarily an open ma …
4
votes
2answers
328 views
The existence of meromorphic functions on Riemann surfaces
In Miranda's book on algebraic curves and Riemann surfaces, Miranda writes:
It is a basic and highly nontrivial
result that a compact Riemann surface
has nonconstant meromo …
1
vote
1answer
78 views
On divisorial correspondences between curves
Assume we are given two smooth curves $C_1$ and $C_2$ over an algebraically closed field $k$. It is known that divisorial correspondences between them correspond to homomorphisms b …
1
vote
0answers
51 views
n-canonical embedding
Let $C$ be a nodal curve and let $ \omega $ be its dualizing sheaf. Let $n$ be a integer larger than 2. Does anyone knows how to show that $\omega^{\otimes n}$ separates points and …
1
vote
1answer
107 views
Trigonal curves of genus three: can their Galois closure be non-abelian
Let $X$ be a curve of genus three which is not hyperelliptic. Then $X$ is trigonal, i.e., there exists a finite morphism $X \to \mathbf P^1$ of degree $3$.
Let $Y\to X \to \mathbf …
12
votes
3answers
671 views
Injective morphism from curves to $\mathbb CP^2$
Is there a smooth compact complex curve that does not admit an injective holomorphic map to $\mathbb CP^2$ ? Let me stress, that the image of the curve in $\mathbb CP^2$ can have s …
0
votes
0answers
52 views
sections of vector bundles transversal to a divisor
Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$.
$E$ a vector bundle over $X$ with a divisor $D$.
We look at the sections $A:=H^{0}(X,E)$ with …
8
votes
1answer
201 views
Injective morphism from an elliptic curve to $\mathbb CP^2$.
Let $E$ be the elliptic curve $x^3+y^3+z^3=0$.
Question. Are there injective morphisms $E\to \mathbb CP^2$ of arbitrary high degree?
Comments. 1) There are injective morphisms $ …

