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8
votes
1answer
252 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
184 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
105 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
619 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
113 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
262 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
69 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
153 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
708 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
175 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
86 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
228 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
281 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
213 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
721 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
811 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
214 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
274 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
106 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
165 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
169 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
282 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
266 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
197 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
172 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
433 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
410 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
207 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 ...
3
votes
1answer
260 views

Igusa invariants of genus 2 curves as Siegel modular functions?

Hi, Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that $\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I_2I_4-3I_6), \quad ...
5
votes
0answers
167 views

Is the moduli space of genus three smooth quartics affine?

Non-hyperelliptic curves of genus three are smooth quartics. Is the moduli space of such curves affine? I think this follows from a more general result on smooth complete intersections, but I'm ...
6
votes
1answer
384 views

If rational points are like entire curves, then what do algebraic points correspond to

I read somewhere that if $X$ is a projective variety of general type over a number field $K$, then rational points are an analogue of entire curves $\mathbf{C}\to X^{an}$ (with $X^{an}$ the ...
3
votes
1answer
270 views

On the m-th power of the Hodge bundle and Arakelov's theorem

Let $S$ be a smooth projective curve over $\mathbf C$ and let $f:X\to S$ be a projective flat morphism with "semi-stable" fibres (i.e., the fibres are reduced and strict normal crossings divisors) and ...
1
vote
1answer
233 views

Hilbert polynomial of $X\times P^1$

Let $X$ be a canonically polarized smooth projective geometrically connected variety over $k$ with Hilbert polynomial $h$. What is the Hilbert polynomial of $X\times_k \mathbf{P}^1_k$? How does it ...
7
votes
1answer
403 views

For which fields does the isogeny theorem hold

Let $k$ be a field. We say that the isogeny theorem holds over $k$ if, for any abelian variety $A$ over $k$, there are only finitely many $k$-isomorphism classes of abelian varieties $B$ over $k$ ...
3
votes
1answer
142 views

Does the Hilbert polynomial determine the weight of the Galois representation associated to a variety

Let $X$ be a curve or an abelian variety (over a finite field). Then the Galois representation associated to $X$ via the etale cohomology of $X$ (in degree $1$) is integral of weight $1$ and its ...
1
vote
2answers
386 views

Equation for simple Jacobian of a genus two curve

Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$. Can we write down an explicit equation for the abelian surface $J$? I know $X$ can be ...
2
votes
1answer
139 views

Detecting sections on an arithmetic variety

Let $S$ be Spec $O_K$ with $O_K$ the ring of integers of a number field $K$. Let $X\to S $ be an arithmetic variety, i.e., an integral smooth quasi-projective $S$-scheme with generic fibre $X_\eta$ ...
3
votes
1answer
215 views

plane cubics and conic bundles

It is well known that any plane cubic curve can be obtained as the discriminant locus of a conic bundle (actually even just of a net of conics). Does this hold true also for all nodal cubics (with ...
1
vote
1answer
193 views

finite surjective morphism to the projective line

Let X a smooth projective curve over $\mathbb{C}$. We fix $d$ distinct closed points $x_{1},\dots,x_{d}$. Can we find a finite surjective morphism $\pi:X\rightarrow\mathbb{P}^{1}$ and local ...
6
votes
2answers
483 views

Questions on theorem in Deligne-Mumford's '69 Paper: $\omega_C^n$ is very ample $n\geq 3$

I'm working through the details of Deligne and Mumford's 69' paper, "The Irreducibility of the Space of Curves of Given Genus", and I had a few quick questions: 1) On p. 77, they claim that for $x$ a ...
6
votes
1answer
268 views

Are ranks of Jacobians over number fields unbounded?

Fix a number field $K$. Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$? Does there exist an integer $g$ such ...
1
vote
1answer
159 views

Books about Stohr-Voloch Theory

Can you suggest any book or lecture notes that explain the theory of Stohr-Voloch? Regards
14
votes
1answer
877 views

Does the moduli space of genus three curves contain a complete genus two curve

Inspired by the question Does the moduli space of smooth curves of genus g contain an elliptic curve and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth ...
2
votes
1answer
336 views

Grothendieck's section conjecture and base change: restricting sections

Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...
7
votes
1answer
282 views

Varieties with infinitely many etale covers and rational points

Let $X$ be a (smooth projective geometrically connected) variety over a field $k$. Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$. ...
22
votes
1answer
929 views

Do all curves have Néron models

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$? By a Néron model, I mean ...
1
vote
1answer
172 views

Constructing a curve with good reduction over a function field

Let $K$ be the function field of a smooth projective connected curve $B$ over $\mathbf{C}$. Let $g\geq 0$ be an integer. Does there exist an nonsingular integral $\mathbf{C}$-scheme $X$ with a ...
2
votes
1answer
309 views

Does a curve over a number field have a finite etale cover of given degree

Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer. Does there exist a curve $Y$ over $K$ with a finite etale ...
1
vote
1answer
248 views

What is the reduction of this hyperelliptic curve

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$. Let $S$ be non-empty finite set of finite places of $K$ and suppose ...
5
votes
1answer
288 views

kapranov's realization of $\overline{M}_{0,n}$ over other fields

Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, or a similar result, ...