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-1
votes
1answer
124 views

effective divisors on a curve and upper semi-continuity

Let consider a smooth projective curve $X$ over $\mathbb{C}$. We consider the scheme that classifies effective divisors of degree $d$, which is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the ...
1
vote
1answer
215 views

a question on the space of divisors on a curve

Let $X$ a complex curve and $x\in X$ a point. We consider the space of effective divisors $D$ with fixed degree $d$, whic we know is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the symmetric group. ...
8
votes
3answers
508 views

What prevents a cover to be Galois?

Let $f:X\rightarrow Y$ be a ramified cover of Riemann surfaces or algebraic curves over $\mathbb{C}$. My question is can one in terms of the ramification data of $f$, determine whether the cover is ...
13
votes
1answer
289 views

Curves which do not dominate other curves

Let $g>1$ be an integer. Does there exist a (smooth projective) genus g curve $X$ which doesn`t dominate a curve of positive genus and genus smaller than $g$? Surely such curves exist. Just take a ...
4
votes
2answers
135 views

Dimension of the space of invariant quadratic differentials in Galois covers

Let $f: X \rightarrow Y $ be a Galois cover of with $X$ and $Y$ algebraic curves over $\mathbb{C}$. I want to compute the dimension of the subspace of $G$-invariants in $H^{0}(X,\omega^{\otimes2})$ ...
5
votes
1answer
419 views

Analogy between Jacobian of curve and Ideal class group

It is excerpt from "Algebraic Geometry Codes Basic ...
2
votes
2answers
108 views

normality of moduli of prym curves

Is the moduli space of Prym curves (curves $C$ with square root of $\mathcal{O}_C$, compactified via admissible covers - by Beauville) of a given genus $g$ normal? Why?
6
votes
1answer
301 views

Kaehler differentials on a nodal curve

Suppose $C$ is an integral nodal curve with one node. It is claimed in the arxiv version of a paper by Bogomolov, Hassett, Tschinkel that the dualizing sheaf and the sheaf of differentials are related ...
8
votes
6answers
775 views

What is a branched Riemann surface with cuts?

Edit: Let me restate the main claim being made in these two papers, Consider the "branched" Riemann surface which has "n" sheets stuck along the intervals, $[z_i, z_{i+1}]$ for $i=1,..,2N$ then it ...
0
votes
1answer
103 views

Is the space of degree $d$ curves with marked smooth points dense inside the space of curves with marked points?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d} $ be the space of nonzero homogeneous degree $d$ polynomials in three variables upto scaling, where $\delta_d = \frac{d(d+3)}{2} $ (basically degree ...
10
votes
1answer
624 views

what is the cyclic cover trick?

What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explaination, both talking about curves and surfaces...
1
vote
2answers
315 views

Existence of non-trivial solution to non linear polynomial system

I need to find conditions for the existence of non-trivial solutions to a multivariable polynomial system in two cases: The first case: $f1: a_1x^2+a_2xy+a_3y^2+a_4z^2=0$ $f2: ...
2
votes
1answer
238 views

explicity equations for curves in the projective space

It is well known that if a smooth curve $C \subset \mathbb{P}^3$ has degree $ d \leq 6$. Then $ g(C) \leq 4$ (Hartshorne pg 354). I know that the case $g=4$ correspond to the complete intersection ...
2
votes
1answer
201 views

finite covering

I am trying to understand the following example, which I came across in a research article. I am posting it as a question below. $\bf{Question}$. Let $\Sigma$ be a curve of genus two with the ...
1
vote
0answers
126 views

What does Hodge theory tell us about simply connected surfaces of general type

Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...
0
votes
2answers
386 views

Recommended books/lecture notes for vector bundle on algebraic curve

I am going to enroll in a ceminar with the topic "vector bundle on algebraic curve". Except Algebraic Geometry(which I think GTM 52 by Hartshone is the main source), which topic I should prepare in ...
3
votes
1answer
383 views

What is the minimal degree of a smooth curves which is not on a cubic surface in $P^3$?

Consider a fixed smooth algebraic curve $C$ over $\mathbb C$. It is well-known that $\mathbb CP^3$ contains curves that are abstractly isomorphic to $C$. What is the minimal degree of a curve in ...
3
votes
1answer
164 views

Families of Hurwitz Curves

Hurwitz's theorem on automorphisms tells us that the group of automorphisms of a nonsingular complex algebraic curve of genus at least 2 is bounded above by $84(g-1)$ where $g$ is the genus of the ...
1
vote
1answer
240 views

$P^1$ minus k points

For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write $$ P^1 \setminus S \cong \mathbb{H}/G $$ where $\mathbb{H}$ is the upper-half plane and $G\subset ...
8
votes
2answers
430 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 trivial, see for instance ...
10
votes
0answers
172 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
206 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 effective line bundle $L$ ...
8
votes
1answer
216 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) $\sigma_1\sigma_2\sigma_3$ is the identity; ...
0
votes
0answers
175 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 (probably twisted) ...
3
votes
0answers
101 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 not? It would be ...
17
votes
2answers
606 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 ramified at every ...
2
votes
0answers
98 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 tangent vectors? ...
1
vote
1answer
234 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 map? That is, is the ...
0
votes
0answers
66 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 $\deg E$ big enough. ...
1
vote
1answer
142 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 between their ...
4
votes
2answers
609 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 meromorphic functions on ...
1
vote
1answer
165 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 $Y\to X \to \mathbf P^1$ be a Galois ...
1
vote
0answers
85 views

normal space of Brill--Noether variety

Let $C$ be a smooth projective curve, $J$ its Jacobian (of degree $d$, parametrizing degree $d$ line bundles, $d \geq 0$). Let $W_d^r$ be the Brill--Noether variety parameterizing degree $d$ line ...
1
vote
1answer
209 views

Picard group of a K3 surface generated by a curve

In Lazarsfeld's article "Brill Noether Petri without degenerations" he mentions the fact that for any integer $g \geq 2$, one may find a K3 surface $X$ and a curve $C$ of genus $g$ on $X$ such that ...
8
votes
1answer
275 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 $E\to \mathbb CP^2$ ...
1
vote
1answer
210 views

Field of definition of a finite etale cover of an anabelian curve

Let $X$ be an anabelian curve over a number field $K$ and let $p:Y\rightarrow X$ be a finite etale cover. Then is anything known (or has anything been conjectured) about the field of definition of ...
5
votes
2answers
637 views

“Arithmetic genus” of a plane curve singularity.

I believe that the following questions are very basic, but I don't know how to get a reference. Consider a curve in the plane $C\in \mathbb C^2$ with a singularity at $0$ and suppose it is ...
13
votes
3answers
794 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 singularities. I ...
0
votes
1answer
200 views

When is a cyclic cover hyperelliptic?

Let us work over the complex numbers for simplicity. Consider a curve $C$ presented as a cyclic cover of some lower genus curve $C'$. When $C'$ has genus $0$, we can write $C$ as the normalization of ...
6
votes
3answers
268 views

Upper bound for the order of the group of automorphisms of Riemann surfaces of genus 2

By the Hurwitz's automorphisms theorem there is an upper bound $|\text{Aut}(C)|\leq 84(g-1)$ for all Riemann surfaces $C$ with $g(C)\geq 2$, but it is not sharp if $g=2$. What is the sharp upper bound ...
0
votes
1answer
104 views

The description of Hurwitz groups

Let $G$ be a Hurwitz group, i.e the automorphism group of some Hurwitz surface $C$. Then Hurwitz's automorphisms theorem shows that the quotient map of $C$ by $G$ has ramification points of indexes ...
2
votes
1answer
161 views

On the group actions on Hurwitz surfaces

Let $C$ be a Hurwitz surface, $G=\text{Aut}(C)$ and $N$ is a proper normal subgroup of $G$. Is there a simple argument (without using of classification theorems) for the fact that $N$ acts on $C$ ...
0
votes
0answers
165 views

sections of vector bundles

Let $X$ a smooth projective connected curve over $\mathbb{C}$. Let $E$ a vector bundle and $E'$ a subbundle of $E$. Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ with $n>2g$ and $z$ ...
3
votes
2answers
272 views

On the construction of the varieties parametrizing special linear series on a curve

Fix an algebraic curve $C$ of genus $g$, and positive integers $d, r$. The variety $W^r_d$ parametrizes complete linear series of degree $d$ and dimension at least $r$ on $C$ and the variety $G^r_d$ ...
3
votes
2answers
262 views

Why the Abel-Jacoby map is algebraic morphism?

The Abel-Jacobi map from the algebraic curve $C$ to its Jacobian $J(C)$ is given analitically by $$p\to \left( \ldots, \int^{p}_{p_0} \omega_i,\ldots\right),$$ where $p_0$ is some point on $C$ and ...
3
votes
0answers
183 views

Stack of vector bundles (on a curve) over a strictly semi-stable point of the moduli space

Consider the stack $Bun_{r,d}^{ss}$ of rank $r$ semi-stable vector bundle of degree $d$ over a fixed curve. There exists also a coarse moduli space $M$ built via GIT. Over the stable locus of $M$ it ...
4
votes
0answers
167 views

Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree

Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer. Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$? I want to exclude finite etale ...
11
votes
1answer
429 views

Complex curves covered by smooth plane curves

Question: Is it true that for every smooth compact complex curve $C$ there exists a smooth curve $C'$ in $\mathbb CP^2$ that admits a non-trivial morphism (i.e. holomorphic map) $C'\to C$? ...
3
votes
1answer
408 views

Name for curve?

I am doing something with the curve given parametrically by $y = (-ar+b) r$, $x = \sqrt{r^2-y^2}$ for $r\in \lbrack (b-1)/a,b/a\rbrack$. It is nice enough (and of low enough degree) that I suspect ...
2
votes
1answer
200 views

Points in the plane imposing independent conditions: reference request

Hello, Does anybody know a reference for the following result: $d\ge 5$ points of $\mathbb P^2$ fail to impose independent conditions on curves of degree $d-3$ if and only if at least $d-1$ of these ...