Questions tagged [algebraic-curves]
for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.
986
questions
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60
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Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles
Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...
- 203
2
votes
0
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211
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What are the Hodge and log Hodge groups of $M_{g,n}$?
I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...
- 1,084
2
votes
1
answer
169
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Lifting of quadrics containing a curve
Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
- 6,000
11
votes
2
answers
755
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What relationship is there between repeated roots of discriminants and orders of roots of the original polynomials?
Disclaimer:
I asked this problem several days ago on MSE, I'm cross-posting it here. The title sounds like a high school problem, but (as a grad student not in algebra) it feels subtle/deep.
...
- 225
0
votes
0
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53
views
Lifting of quadrics containing hyperplane section for projectively normal curves
Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
- 6,000
4
votes
1
answer
246
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Intersection complex of genus-zero curves?
I would like to have a very explicit description of $\bar M_{0, n}$, especially its boundary divisors and how they intersect. All I can do in my construction is add divisors and blow up at strata, ...
- 1,084
2
votes
1
answer
285
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Existence of curves of a given degree in threefolds
Let $X$ be a projective complex smooth threefold such that its Picard group is generated by an ample line bundle $L$. I have the following question:
For each given integer $d\geq 1$, does there exist ...
- 111
1
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0
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73
views
Unique polarization on a very general curve with Mumford-Tate
I try to understand why a very general curve (smooth, projective over $\mathbb{C}$) has an unique polarization up to scalar on the $H^1(X,\mathbb{Q})$.
I was advised to look at the maximality of the ...
6
votes
0
answers
157
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Does there exist a plane curve such that it has the heart curve as catacaustic?
Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$.
The catacaustic ...
- 545
7
votes
1
answer
245
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Computing $\pi_1$ of the complement of a non-singular plane curve
The following is a well-known fact:
Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$.
This was further ...
- 10.5k
1
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0
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107
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Number of conditions imposed by general points
I encountered with a problem when I read the part of Enriques-Babbage Theorem of the book Geometry of Algebraic Curves Vol. I by ACGH. It is stated on page 112-113 that all subsets of $m$ points of a ...
- 393
3
votes
1
answer
232
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Symmetric differential forms on moduli space of curves
Do there exist regular symmetric differential forms on $\overline{\mathcal{M}}_{g,n}$ the DM-stack of stable genus $g$ curves with $n$ marked points? By this, I mean nonzero sections $$ \omega \in H^0(...
- 3,301
2
votes
2
answers
270
views
Involution of the symmetric square of a smooth plane quartic
Let $C$ be a smooth plane quartic defined over a field $K$. Denote by $J$ its Jacobian, and by $C^{(2)}$ its symmetric square. Since $C$ is a smooth plane quartic, it is non-hyperelliptic, and hence ...
- 1,641
2
votes
1
answer
125
views
Endomorphism ring of the Jacobian of a generic smooth plane quartic
Let $C$ be an arbitrary smooth plane quartic defined over a number field $K$. Assume $C$ is not hyperelliptic, and denote by $J$ the Jacobian of $C$. How does $\text{End}(J)$ look like for a generic ...
- 1,641
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votes
1
answer
184
views
Explicit description of dualizing sheaf of nodal curve
Let $C$ be a nodal curve with one single node $p$ and $f: N \to C$ it's normalization. Let $r,s $ preimages of $p$. In Geometry of Algebraic Curves (p 91) is stated without proof that the dualizing ...
- 6,000
5
votes
2
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230
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Characterize the space of all ramification divisors of degree $d$
Let $X$ be a compact Riemann surface of genus $g>0$, and let $f\colon X \to \mathbb{P}^1$ be a branched covering of degree $d$. Define the ramification divisor $R_f$ on $X$ by $f$, where $\deg R_f =...
- 713
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0
answers
142
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Inclusion of boundary strata of moduli of curves: induced map on tangent spaces
$\DeclareMathOperator\Ext{Ext}$Let $C \in \bar{\mathcal{M}}_g$ be a nodal curve. It is a classical result that the tangent space of $\bar{\mathcal{M}}_g$ at $C$ is given by
\begin{align*}
T_C \bar{\...
- 203
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votes
1
answer
203
views
Stable curve local complete intersection
Let $C$ be a stable curve over base field $k$. How to show that $C$ is local complete intersection purely algebraically?
I'm emphasizing pure algebraically here because the only proof of this ...
- 6,000
7
votes
1
answer
256
views
How to construct such a real algebraic curve
Suppose $L$ is a real line in $\mathbb{RP}^2$, my question is: for a given postive integer $n$, is it possible to find a real projective algebraic curve $C$ of degree $n$ with maximum connected ...
- 85
1
vote
0
answers
133
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Calculate genus of reducible nodal curve
Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are ...
- 6,000
4
votes
1
answer
175
views
Gonality of specific Riemann surfaces $y^k=\tfrac{z^k-1}{z^k+1}$
The gonality of a compact Riemann surface $\Sigma$ is defined to be the lowest degree $d$ of a non-constant holomorphic map $f\colon \Sigma\to\mathbb CP^1.$ This means the gonality is 1 only for $\...
- 6,735
1
vote
0
answers
149
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A quick introduction to the birational classification of projective curves
To give you some personal background: I am a ring theorist, and most of my research focus on invariant theory of noncommutative rings. Recently I became interested in a certain problem that requires a ...
- 2,683
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votes
0
answers
69
views
Is there an $\mathbb{F}_{\!q}$-curve of geometric genus 3 and $\mathbb{F}_{\!q^3}$-cover to an elliptic $\mathbb{F}_{\!q}$-curve of $j$-invariant 0?
Let $E$ be an elliptic curve $y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $3 \mid (q-1)$. Is there an absolutely irreducible $\mathbb{F}_{\!q}$-curve $C$ of ...
- 1,801
1
vote
0
answers
113
views
Singularity of curves on very general surfaces
I want to ask if there is a known classification of possible singularities of curves on a general (or very general) surface in $\mathbb{P}^3$.
It was shown in Proposition 3 of "Subvarieties of ...
- 19
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votes
1
answer
251
views
Is there an isotrivial elliptic surface of positive rank having a section of order $3$?
Let $k$ be a field of characteristic $p > 3$. I cannot find any example of ordinary isotrivial elliptic $k$-surface $E$ (i.e., elliptic $k(t)$-curve, where $t$ is a variable) whose Mordell-Weil ...
- 1,801
6
votes
1
answer
535
views
Is every smooth projective curve a modular curve?
I have seen a quote saying that
Every smooth projective curve over a number field is a modular curve, i.e. (compactification of) $\Gamma\backslash\mathcal{H}$ for some finite index subgroup $\Gamma&...
- 1,378
1
vote
1
answer
206
views
The ideal in $\mathbb{Z}[x]$ of all vanishing polynomials of a curve automorphism
Let $C$ be a projective irreducible algebraic curve of genus $g$ over an algebraically closed field of characteristic $0$ (for simplicity). Given an automorphism $\alpha \in \mathrm{Aut}(C)$ of order $...
- 1,801
2
votes
1
answer
282
views
One unexpected observation related to algebraic curves and their Jacobians
Let $C$ be a projective irreducible algebraic curve over an algebraically closed field of characteristic $0$ (for simplicity). Assume that there is also a cover $\varphi\!: C \to E$ to an elliptic ...
- 1,801
0
votes
1
answer
262
views
Any irreducible projective curve in $\mathbb P^3$ can be defined by three functions
This question is asked in MSE but no effective answer appeared.
Suppose $C$ is a irreducible closed curve in $\mathbb P^3$(projective space over an algebraically closed field), I need to prove there ...
- 523
2
votes
1
answer
232
views
(Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve
Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...
- 1,801
1
vote
0
answers
77
views
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$ ...
- 1,641
0
votes
1
answer
141
views
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 ...
- 1,641
1
vote
0
answers
78
views
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,
$$
where $a_1(x), \dots, a_n(x)$ are ...
- 89
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votes
0
answers
129
views
Integration on algebraic curves
Consider the plane algebraic curve
$$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 ...
- 89
14
votes
1
answer
349
views
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 ...
- 489
1
vote
0
answers
107
views
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 ...
- 11
1
vote
0
answers
139
views
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 ...
- 469
7
votes
0
answers
259
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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)) = \...
- 131
6
votes
2
answers
399
views
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 ...
- 595
1
vote
0
answers
83
views
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 ...
- 595
0
votes
0
answers
83
views
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(\...
- 25.5k
4
votes
1
answer
241
views
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 \...
- 1,701
3
votes
0
answers
120
views
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 ...
- 31
3
votes
1
answer
207
views
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 ...
- 2,195
1
vote
0
answers
90
views
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 $...
- 1,801
1
vote
0
answers
130
views
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, ...
- 595
3
votes
0
answers
176
views
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 ...
- 1,801
5
votes
1
answer
239
views
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 ...
- 197
1
vote
0
answers
179
views
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,...
- 3,301
3
votes
1
answer
277
views
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(\...
- 2,195