**3**

votes

**2**answers

462 views

### Is a normal proper relative curve over a DVR projective?

Let $X$ be a connected normal scheme equipped with a proper flat morphism $f\colon X \rightarrow \mathrm{Spec }(R)$ with $R$ a discrete valuation ring and such that the fibers of $f$ are curves (i.e., ...

**6**

votes

**1**answer

214 views

### Evaluation maps for moduli of stable maps

Let $\overline{M}_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$.
Consider the product of the evaluation ...

**5**

votes

**2**answers

246 views

### Disjoint curves in an algebraic surface

Let $X$ be an algebraic surface (over the complex) with $p_g=q=0$. Is it possible to have disjoint curves $C_1,\ldots, C_b$, of positive genus, spanning $H_2(X,{\mathbb Q})$, $b=b_2(X)$?
(When $X$ is ...

**3**

votes

**1**answer

315 views

### On equations defining space curves

I am reading a text by Prof. Szpiro Tata lectures on equations defining space curves. In the proof of Proposition $1.2$ on page $12$ he gives explicit description of the defining equations of a local ...

**2**

votes

**1**answer

157 views

### Frey's Formula and utilisation of the Hasse Invariant in “Links between Stable elliptic curves and Diophantine equations.”

In the paper "Links between Stable elliptic curves and Diophantine equations" for an elliptic curve $E$ with normal Weierstrass form $$y^2 = x^3 -g_2x -g_3$$ with $g_i \in \mathbb{Z}$ w.l.o.g. Then ...

**5**

votes

**2**answers

389 views

### Obstruction and rational points on curves

Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?

**7**

votes

**0**answers

245 views

### Map of the Klein quartic from $CP^2$ to $R^3$

The Klein quartic $\mathcal{Q}$ is cut out of $\mathbb{CP}^2$ by the homogeneous equation $$x^3 y + y^3 z + z^3 x = 0.$$ It has 168 orientation preserving automorphisms and includes several copies of ...

**1**

vote

**1**answer

188 views

### Degree of irreducible locally free sheaves and global sections on curves

Let $X$ be a smooth projective curve and $\mathcal{F}$ a locally free sheaf on $X$ of rank $2$ and negative degree. Assume further that $\mathcal{F}$ is irreducible in the sense, $\mathcal{F}$ cannot ...

**8**

votes

**2**answers

649 views

### Sum of consecutive cubes

I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does
$$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$
have nontrivial solutions $(k,...

**5**

votes

**0**answers

155 views

### Is the moduli space of curves arising from wild ramification smooth?

Fix a natural number $g$, a prime $p$, and a $p$-group $P$.
Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...

**1**

vote

**1**answer

196 views

### Relation between intersection and product of ideals

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...

**0**

votes

**1**answer

185 views

### automorphism group of a function field

Suppose that F is a function field of a single variable over a finite field. The automorphism group Aut(F) acts on the places of F and permutes all places of a given degree. I have a few questions:
1)...

**1**

vote

**0**answers

91 views

### Curve associated to bipartite graph

Given real biadjacency matrix $A\in\{0,1\}^{n\times n}$ of a bipartite graph with rank $r\in[2,n-1]$, denote $A(x)$ to be matrix where $0$ is replaced by $x$ and $1$ by $1-x$. Denote $$p_1(t,x)=Det(tI-...

**1**

vote

**0**answers

90 views

### Normalization (integral closure) of $\mathbb Z_p[x]$ in function field of a curve to obtain Model of curve

I want to follow this construction of a normal model of a curve:
Let $p\neq 2,3$ and $Y\to \mathbb P¹$ be a smooth projective curve over $\mathbb Q_p$ with function field $L/\mathbb Q_p(x)$ e.g. $L=\...

**9**

votes

**1**answer

386 views

### Does a semistable curve descend to a regular base?

Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that
$f$ is proper, flat, and of finite presentation;
...

**3**

votes

**1**answer

243 views

### When does a hyperelliptic Riemann surface admit a map of degree 3

Let $X$ be a hyperelliptic curve of genus $g>1$.
For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$?
I think a genus two curve $X$ admits a map of degree $3$.
Proof: Pick $P$ ...

**0**

votes

**1**answer

143 views

### Schematic image of a relative Cartier divisor of a fiberwise dense open

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...

**2**

votes

**2**answers

294 views

### Rigid curves, and the “richness” of their homology class

Let $X$ be a complex smooth projective variety, and $C\subset X$ a smooth curve. Then $C$ defines a cycle $$\beta=[C]\in H_2(X,\mathbb Z).$$
I have a very vague question about this situation:
Q. ...

**3**

votes

**0**answers

221 views

### Symmetric power of an etale map of curves

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th ...

**4**

votes

**1**answer

217 views

### Jacobian of a semistable curve

My question is about the proof of Example 8 in section 9.2 of the book "Neron models." There we have a semistable curve $X$ over an algebraically closed field $K$ and we let $\pi\colon \widetilde{X} \...

**1**

vote

**1**answer

264 views

### Moving a divisor on a (reducible, non-reduced) curve

I am trying to understand the first sentence of the proof of 9.1/5 in "Neron models." There we have a proper curve $X$ over a field $K$ and a line bundle $\mathscr{L}$ on $X$. Our ultimate goal is to ...

**3**

votes

**0**answers

352 views

### A Step in the Proof of the Drinfeld-Simpson theorem

I hope that this is the appropriate place for asking about a step I don't understand in a proof which I think is due to a lack of knowledge. This is a step in Drinfeld-Simpson's paper: ``$B$ ...

**6**

votes

**1**answer

221 views

### The cohomology of an $S_{3}$ cover of an elliptic curve ramified in one point

Let $E/\mathbb{C}$ be an elliptic curve. Let $C \to E$ be a Galois cover with group $G = S_{3}$ (symmetric group on $3$ elements), ramified in one point. (To clarify: there is a unique point in $E$ ...

**-1**

votes

**1**answer

57 views

### How to prove embedded copies of a curve using different base points in its Jacobian are algebraically equivalent

Let $X$ be a smooth projective curve over $k\subset\mathbb{C}$, and $p,q\in X(k)$. Let $X_p$ (resp. $X_q$) be the embedded copy of $X$ in the Jacobian $Jac(X)$ using the base point $p$ (resp. $q$). Is ...

**0**

votes

**0**answers

133 views

### Chow group of a product

Let $X$ and $Y$ be smooth varieties over $k$. I was wondering if there is a decomposition of the Chow group $CH(X\times Y)$ in terms of $CH(X)$ and $CH(Y)$ similar to the Kunneth decomposition of $H(X\...

**0**

votes

**3**answers

486 views

### Why there are two point at infinity on certain elliptic curve [closed]

In article Adams, W. W., & Razar, M. J. (1980). Multiples of points on elliptic curves and continued fractions. Proc. London Math. Soc, 41, 481-498. is said on ...

**9**

votes

**1**answer

305 views

### Rationality of moduli spaces of rational curves

Let $\overline{M}_{0,n}$ be the moduli space of Deligne-Mumford stable pointed rational curves, and let us consider the quotient $\widetilde{M}_{0,n} = \overline{M}_{0,n}/S_n$.
Clearly, there is a ...

**1**

vote

**1**answer

160 views

### About Prop 1.1 of “…Petri's Analysis…” by Stöhr & Viana

I'm reading this paper and the authors Karl-Otto Stöhr and Paulo Viana define $\Omega^n$ to be the set of the sum of the monomios $\omega_1\cdots\omega_n$, where $\omega_i\in \Omega$ and $\Omega^n(D)$ ...

**2**

votes

**0**answers

73 views

### Locus of line bundles with given base points on a curve

Let $C$ be a smooth projective curve over an algebraically closed field. If $D$ is an effective divisor on $C$ (let's say reduced to make things easier) of degree $m$ and $d>m$, is the dimension of ...

**2**

votes

**0**answers

95 views

### Abel-Prym map for Prym-Tyurin varieties

Let $(J,\Theta)$ be the Jacobian of a smooth projective curve $C$, and let $i:P\hookrightarrow J$ be an abelian subvariety of $J$ such that $i^*\Theta\equiv e\Xi$ for some principal polarization $\Xi$ ...

**8**

votes

**4**answers

423 views

### Is there a non-abelian version of the Torelli map?

Let $C$ be a connected compact oriented real surface of genus $g$, let $G$ be a connected compact Lie group and let $G_\mathbb{C}$ be the complexification of $G$. One considers the moduli space $M (C,...

**6**

votes

**0**answers

281 views

### Global sections for a locally free sheaf over curves

Let $B$ be a complete algbraic curve of genus $g$, and $\mathcal{E}$ be a semi-stable locally free sheaf of rank $r$ over $B$. Assume that the slope of $\mathcal{E}$ is $\mu(\mathcal E):=\frac{\deg \...

**3**

votes

**1**answer

178 views

### Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$

Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = \mathrm{PSL}_2(\...

**2**

votes

**0**answers

60 views

### Finite extension of K(x) with extra structure: definable over field of invariants?

Let $K$ be an algebraically closed field, and let $\sigma$ be an automorphism on $K$. Set $k=K^\sigma$. Consider the rational function field $K(x)$ and extend $\sigma$ to $K(x)$ by $\sigma(x)=x$, ...

**2**

votes

**2**answers

413 views

### What is the difference between the moduli space of curves and the moduli space of orbi-curves?

Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write.
I feel that I should already know the answer to this, but it never sits ...

**4**

votes

**2**answers

193 views

### Map between stacks and automorphism groups

I know that the Torelli morphism $t_g:\mathcal{M}_g\rightarrow \mathcal{A}_g$ between the stacks of smooth curves of genus $g$ and principally polarized abelian varieties of dimension $g$ is of order ...

**5**

votes

**0**answers

117 views

### Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind.
Let $X$ be a smooth quasi-projective geometrically ...

**2**

votes

**1**answer

311 views

### On a property of the Grothendieck group of a smooth projective curve

Let $K$ be a complete DVR of characteristic $0$, $X$ a smooth projective curve over $K$. Denote by $K^0(X)$ the Grothendieck group of locally free sheaves on $X$ and by $\mbox{det}$ the natural group ...

**3**

votes

**0**answers

91 views

### Criterion for the existence of finite locally free resolution

Let $X$ be a projective variety over an algebraically closed field $k$, $S$ be a $k$-scheme, $E$ be a coherent sheaf on $X \times_k S$, flat over $S$. We know that if $X$ is smooth then $E$ has a ...

**6**

votes

**2**answers

260 views

### Non trivial family of hyperelliptic curves

Let $X$ ba a smooth hyperelliptic curve of genus $g$, and let $f:X\rightarrow X$ be the hyperelliptic involution. Consider a $K3$ surface $S$ with an involution $g$ without fixed points. The quotient $...

**2**

votes

**1**answer

241 views

### discriminant of smooth quartic del Pezzo surface in $\mathbb{P}^4$

I can't understand the proof of Lemma3.3 in Stability of genus 5 canonical curves.
Let $C$ be a complete intersection of three quadrics in $\mathbb{P}^4$ and let $\Lambda$ be the net of quadrics ...

**5**

votes

**1**answer

154 views

### p-adic L-function of curves

Given a smooth projective curve $C$ over $\mathbb{Q}$ one has the $L$-function $L(C, s)$ and the Beilinson conjectures predict its values at integers $s=n$ in terms of regulators.
Is there a p-adic ...

**1**

vote

**0**answers

101 views

### plane curves with two points of high multiplicity

Let $\mathcal{C}$ be an irreducible plane curve in $\mathbb{P}^2_\mathbb{C}$ of degree $d$. Let $D$ be a quartic with three irreducible components with normal crossing singularities, i.e. a conic and ...

**0**

votes

**1**answer

124 views

### Is the support of two odd theta characteristics on a generic curve disjoint?

Concise version of the question
On a generic curve of genus $g$, the odd theta characteristics will have exactly one global section. Therefore each odd theta characteristic will correspond to a ...

**7**

votes

**1**answer

557 views

### What are the exact holomorphic Lagrangians in complex 2-space?

In an exact symplectic manifold, i.e. where the symplectic form can be written $\omega = d \lambda$, it's natural to look for exact Lagrangians, i.e. $L$ on which $\lambda_L = df$. One reason is ...

**2**

votes

**1**answer

183 views

### curve through a point avoiding an hypersurface, II

Inspired by this question:
Suppose given an algebraic curve $C \subset \mathbb{A}^2$, and a point $x \in C$. Can you find another (closed) curve $D \subset \mathbb{A}^2$ such that $C \cap D = x$?

**1**

vote

**2**answers

179 views

### Jacobian of a curve and field extension

Let $K$ be a field of characteristic zero and $X_K$ a smooth projective curve on $K$. Denote by $\bar{K}$ the algebraic closure of $K$ and $X_{\bar{K}}$ the base change of $X_K$ to $\bar{K}$. Under ...

**3**

votes

**2**answers

303 views

### Injectivity under flat base change of the Picard group on smooth projective curves

Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...

**6**

votes

**1**answer

303 views

### Realizing algebraic curves as complete intersections

I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$).
Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in $...

**10**

votes

**1**answer

676 views

### Why is the section conjecture important?

As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...