# Questions tagged [algebraic-curves]

for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.

959
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### Smooth proper local model of a smooth projective curve

Say I have a curve $C/K$, where $K$ is a number field. Let $v$ be a place of $K$, and denote by $K_v$ the $v$-adic completion of $K$. Further assume $C$ is smooth and proper over $K$. Denote by $C_v$ ...

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### Smooth proper models of hyperbolic curves

Let $C$ be a curve defined over a number field $K$. I am interested in knowing under which conditions on $C$, does it have a smooth proper model?
I understand that when $C$ is a smooth, projective, ...

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### Properties of the symmetric square of a curve [migrated]

Under what conditions on a curve $C$, defined over a ring $R$, is its symmetric square, $C^{(2)}$ smooth/proper? Is it enough for $C$ itself to be smooth/proper over $\text{Spec}R$?
How does one see ...

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1
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115
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### Embedding of symmetric square in Jacobian

Let $C$ be a projective curve defined over a field $K$, and let $C^{(2)}$ and $J$ be its symmetric square and Jacobian, respectively.
There is a natural map $C^{(2)}\hookrightarrow J$, defined as ...

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### Canonical basis of cycles of Riemann surfaces

Let $\Gamma$ be the compact Riemann surface defiend by the algebraic curve
$$
f(x, y) = y^n + a_1(x)y^{n-1} + a_2(x) y^{n-2} + \dots + a_{n-1}(x)y + a_n(x) = 0,
$$
where $a_1(x), \dots, a_n(x)$ are ...

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111
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### Integration on algebraic curves

Consider the plane algebraic curve
$$f(x, y) = y^4 - (2x - 1)y^2 - (4x - 1) y + x^2 + x + 1 = 0.\tag{1}$$
Its compactification results in a Riemann surface $C_1$ of genus $1$.
Hence, it can be ...

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### Existence of space curves of given genus and degree

In Hartshorne's Algebraic Geometry Chapter IV, Section 6, he summarizes known results on the existence of smooth space curves of degree $d$ and genus $g$ for $g\le 12$ and $d \le 10$. He shows the ...

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### Degeneration differential form nodal curve

I have a (possibly very basic) question about differential forms on nodal curves. After reading Witten's survey "Two-dimensional gravity and intersection theory on moduli space", I am ...

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117
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### Representation of automorphism group of a curve acting on points of finite order in the Jacobian

Let $C$ be a curve of large genus $g > 1$ over an algebraically closed field of characteristic $0$, and let $G = \textrm{Aut}(C)$ be its automorphism group. Is there a general way to compute the ...

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### Complete curves in $\mathcal{M}_g$ all of whose Jacobians have trivial endomorpism ring

I'm trying to construct complete smooth curves $C$ in $\overline{M}_g$ such that for all points $S \in C \cap \mathcal{M}_g$, its Jacobian $\text{Jac}(S)$ satisfies $\text{End}(\text{Jac}(S)) = \...

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363
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### Good and bad reduction for twists of algebraic curves

Suppose we have two curves $C/\mathbb{Q}$ and $C'/\mathbb{Q}$ which are twists of each other i.e. they are isomorphic over a field extension $K/\mathbb{Q}$.
Suppose that $C$ has good reduction at a ...

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72
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### Bad primes of twists of modular curves $X_E^{-1}(p)$

I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...

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### On the distances from an algebraic curve to integer points

Let $C$ be an algebraic plane curve, considered as a subset of $\mathbb{R}^2$, defined over $\mathbb{Q}$. For each $\vec x \in \mathbb{Z}^2 \setminus (C \cap \mathbb{Z}^2)$ define
$$\displaystyle D(\...

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### Irrational Fano threefold whose intermediate Jacobian is Jacobian of curve

Clemens-Griffiths criterion for 3-fold says that if a smooth projective threefold $X/\mathbb C$ is rational, then the intermediate Jacobian $J(X)$ is isomorphic to product of Jacobians $J(C_1)\times \...

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### How many elliptic curves over a finite field have a square discriminant?

$\newcommand{\char}{\operatorname{char}}$Given a finite field $F_q$ with $q\equiv 1 \bmod 3$ and $\char(F_q)>3$, I need to figure out how many isomorphism classes of elliptic curves $E/F_q$ have a ...

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### Degeneration of curves in smooth families

Heuristically, I want to know, given a smooth, projective morphism from a scheme to a discrete valuation ring, if the generic fiber can be 'covered' by a family of geometrically integral curves, is it ...

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### Is it possible to lift a pair of points on an elliptic $\mathbb{F}_{\!q}$-curve to a pair of short points on an elliptic $\mathbb{F}_{\!q}(t)$-curve?

Let $E$ be an (ordinary) elliptic curve over a finite field $\mathbb{F}_{\!q}$ of a (quite large) characteristic. For simplicity, suppose that $E(\mathbb{F}_{\!q})$ is a prime group. In addition, let $...

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### Supersingular points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ and their Frobenius action

I am trying to understand certain points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ (where $p$ is a rational prime) via their moduli description.
A point $P \in X_0(p)$ can be viewed as $P:=(E, ...

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### Does every (ordinary) elliptic curve over a quite large prime field $\mathbb{F}_{p}$ have a lift to the function field with huge Mordell-Weil rank?

Ulmer (and then other authors) showed existence of elliptic curves $E$ over the function field $\mathbb{F}_{p}(t)$ (where $p$ is a prime) with huge Mordell-Weil ranks (i.e., depending on $p$). The ...

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### Complement of plane curve and knot

In Libgober's paper Alexander polynomial of plane algebraic curves and cyclic multiple planes, Example 2 (p.850), Libgober claims that the complement to this curve (i.e. $x^2u=y^3$ relative to the ...

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### Does inequality on arithmetic genera hold for all normal models of curves

Let $f : X \to \mathrm{Spec}(R)$ be a model of a curve. Explicitly, f is flat, proper of relative dimension 1, and R is a dvr with fraction field K and residue field k of characteristic p. Furthermore,...

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### Segre embedding and intersections by hyperplanes

Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...

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### A question about algebraic indicator functions

Let $f \in \mathbb{Z}[x]$ and $m,k \in \mathbb{Z}$. Consider the indicator function $g_f : \mathbb{Z} \to \{1,0\}$ given by
\begin{align*}
g_f(n) =
\begin{cases}
1 &\text{if there exists $r \in \...

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134
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### Question about algebraic curve being birational to smooth projective curve

Let $X$ be a geometrically irreducible affine variety defined over $\mathbb{Q}$ and dimension $1$. Then it is known that $X$ is birational over $\mathbb{C}$ to a smooth projective curve $C$.
I was ...

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### Writing a smooth plane quartic as the vanishing of $Q_0Q_2 - Q_1^2$ for quadratic $Q_0,Q_1,Q_2$

It seems well-known that any smooth plane quartic can be written as the vanishing of $Q_0Q_2 -Q_1^2$. Is there a good way to work out these quadratic factors $Q_0,Q_1,Q_2$? For example, given the ...

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### Are maps into a smooth curve equivalent to relative effective Cartier divisors?

Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...

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### Degree of the syzygy bundle of a curve of genus 3

Let X be an hyperelliptic curve of genus 3, 𝜔 its canonical sheaf, and M the syzygy bundle of 𝜔.
What is the degree of M?

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### What are algebroid curves/branches and their value semigroup?

In “The moduli problem for plane branches”, by O. Zariski, the author defines a plane branch as an irreducible element $f \in \mathbb C[[x,y]]$. In the more recent article "The semigroup of a ...

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### Matrix description for automorphisms of genus $2$ curve split into two copies of an elliptic curve

Consider a genus $2$ curve $C$ and its Jacobian $J$ (for simplicity, over the field $\overline{\mathbb{Q}}$ or $\mathbb{C}$). Assume that $J$ is $(2,2)$-isogenous to the direct square $E^2$ of an ...

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### Global sections of relative characteristic of log-smooth curves

$\DeclareMathOperator\Spec{Spec}$I am currently learning about log-geometry and try to understand the theory in the example of curves with basic log-structure over a general base $S$. Especially, I am ...

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### Finding an $\mathbb{F}_q$-point on one specific intersection of quadrics

Let $\mathbb{F}_q$ be a finite field of large characteristic and $a_1, a_2, \cdots, a_n \in \mathbb{F}_q$ be some pairwise different elements. I assume that $\sqrt{-1} \in \mathbb{F}_q$. Consider the ...

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### Finiteness of automorphism group of finite map $f: C \to \mathbb{P}^1$

Let $C$ be a connected curve of arithmetic genus $g$
over algebraically closed field $k$ of characteristic zero having only nodes
as singularities together with finite morphism
$f: C \to \mathbb{P}^1$....

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1
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### Image of $H^0(C,\omega_C-x)$ in $G(g-1,H^0(C,\omega_C))$

Let $C$ be an algebraic curve over $\mathbb{C}$ and $\omega_C$ be its canonical bundle. We may assume that $C$ has genus $g\geq2$. Let $x\in C$ be an arbitrary point.
Question: What is the image of $...

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### What is the algebra structure on the pushforward of the structure sheaf along a finite map to $\mathbb{P}^1$?

$\newcommand{\P}{\mathbb{P}}\newcommand{\O}{\mathcal{O}}$ Let $f : C \to \P^1$ be a ramified finite map of degree $d$ of smooth algebraic curves over an algebraically closed field $k$. How can we ...

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### What do we know about the $\mathbb{F}_{q^2}$-curve $X^{q-1} + Y^{q-1} + Z^{q-1}$?

People have thoroughly studied the Hermitian $\mathbb{F}_{q^2}$-curve $H: X^{q+1} + Y^{q+1} + Z^{q+1} = 0$. What do we know about the similar $\mathbb{F}_{q^2}$-curve $C: X^{q-1} + Y^{q-1} + Z^{q-1} = ...

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### Equivalent characterizations of rational normal curve

A rational normal curve $C \subset \mathbb{P}_k^d$ (assume $k= \mathbb{C}$) can be defined usually up to projective equivalence in two equivalent ways:
smooth irreducible nondegenerate curve $C \...

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### The $H^1$ of a smooth curve and its (generalized) Jacobian variety

Let $C$ be a smooth projective curve of genus $\geq 1$ over a number field $k$ with a $k$-rational divisor of degree $1$ inducing the embedding $C \hookrightarrow J$, where $J$ is the Jacobian variety ...

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### Is it possible to define the generalized Jacobian of a curve when the modulus $\mathfrak{m}$ is supported on points of higher degree?

The book Algebraic groups and class fields by Serre explains a lot about the construction of the generalized Jacobian $J_\mathfrak{m}$ of a smooth projective curve $X$ with respect to the modulus $\...

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### Motivation of Zariski–Van Kampen theorem

The Zariski–Van Kampen theorem gives the presentation of the fundamental group of the complement of the plane curve of degree $d$. But what's the motivation of this theorem? More generally, why are ...

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### Vanishing odd theta characteristics on plane curves

Are there, for any $k$, smooth plane curves $C\subset\mathbb{P}^2$ of degree $d=2k$ over $\mathbb{C}$ such that the space of global sections of all odd theta characteristics on $C$ is one dimensional?
...

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### Fibrations of curves whose singular locus on the base is not codimension $1$

Let $f : X \to B$ a relative curve meaning a flat proper map whose fibers are geometrically connected $1$-dimensional schemes. In what follows,let $B$ be a smooth variety over $\mathbb{C}$ and $f$ be ...

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### If a genus 2 curve has no $k$-rational points, can it have a $k$-rational divisor class of degree 1?

Let $C$ be a smooth projective geometrically connected curve of genus 2 defined over a number field $k$. Here are some definitions:
The index $I$ of a curve $C$ is the greatest common divisor of all ...

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### Extension of the trivial bundle by the canonical bundle on a curve

Let $X$ be a smooth projective curve over a field $k$ and $K_X$ be its canonical line bundle. By the Serre duality, $\text{H}^1(X,K_X)$ is a one-dimensional $k$-vector space. On the other hand, $\text{...

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### Density of two algebraic curves with maximum number of intersection points

Let $f$ and $g$ be two complex polynomials of degree $n$ and $m$ with two variables respectively and coefficients of these two polynomials correspond to $M=\mathbb{C}^\frac{(n+1)(n+2)+(m+1)(m+2)}{2}$ ...

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### Prescribed intersection of varieties

Every variety here is complex analytic, or complex algebraic if it solves anything.
Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...

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102
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### For curves $C$ of genus $1$, the period (or index?) of $C$ is greater than $1$ iff $C(k)$ is empty

As the title suggests, does anyone have a reference for the proof of this fact? Actually, I can't remember where I've seen it before, or if I even remembered the statement correctly. Here are some ...

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3
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522
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### On the Klein quartic and the similar $a^2b+b^2c+c^2a$?

Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}$.
I. Degree 5
\begin{align}
a &= q^{11/...

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483
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### Hypersurface of singular plane cubics

In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The ...

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### Counterexample to purity of Brauer group for curves

The purity of Brauer group states that for a smooth (quasi-)projective variety $X$ over a field $k$, removing a closed subscheme $Z \subset X$ of codimension at least $2$ ensures that the restriction ...

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### Invariance of numerical class of a curve in Higgs-Grassmann schemes

Premise
Let $X$ be a projective variety of dimension $n\geq1$ over an algebraically closed field of characteristic $0$.
A Higgs sheaf $\mathfrak{E}$ is a pair $(E,\varphi)$ where $E$ is a $\mathcal{...