Questions tagged [algebraic-curves]
for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.
1,059 questions
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Legendrian curves in $\mathbb{P}^3_{\mathbb C}$ with high genus relative to degree
Let $C \subset \mathbb{P}^3$ be a smooth (complex) Legendrian curve of genus $g$ and degree $d$.
(The contact structure on $\mathbb{P}^3_{\mathbb C}$ is unique up to projective automorphism.)
Question ...
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234
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Barton - Kleiman Criterion for the ampleness is sharp
Does there exist a smooth complex projective variety $X$ of dimension $n\geq2$, a rank $r\geq2$ vector bundle $E$ over $X$ such that
$\det(E)$ is ample;
for any smooth complex projective curve $C$, ...
- 990
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255
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Understanding Nagata's 1959 proof of his conjecture for large square
In his 1959 paper On the 14-th problem of Hilbert, Nagata proved his conjecture when $r=s^2\geq16$. I am confused by the "obvious" Lemma 5 of the paper
Lemma 5. If $C$ is a plane curve of ...
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113
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Curve with ordinary cusps on an algebraic surfaces
Let $X$ be a smooth algebraic surface and let $L$ be a very ample divisor on it.
Question. Under which conditions on the pair $(X, \, L)$ can we find a curve $C \in |6L|$ having $6L^2$ ordinary cusps ...
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For a smooth projective curve $X$, is the map, $M(r,d)\rightarrow \mathrm{Pic}^d(X)\cong \mathrm{Pic}^0(X)$ an albanese morphism?
$\DeclareMathOperator\Pic{Pic}$Let $X$ be a smooth projective curve and $M(r,d)$ be the moduli space of semistable vector bundles of rank $r$ and degree $d$ over $X$. My aim is to show that, when we ...
- 193
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Set theoretic complete intersection property of an algebraic curve
I am working on a research problem and would like clarification on the following question.
Let $V$ be an irreducible algebraic curve in $\mathbb{C}^d$ with $d>3$. Suppose $V$ is contained in an ...
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Do you know an (explicit) example of a superelliptic curve $C\!: y^{11} = f(x)$ for which there is a cover $C \to E$ onto an elliptic curve $E$?
Do you know an (explicit) example of a superelliptic curve $C\!: y^{p} = f(x)$ over a field of zero characteristic for which $p = 11$ and there is a cover $\phi\!: C \to E$ onto an elliptic curve $E$? ...
- 1,873
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140
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Embedding plane quintics in $\mathbb{P}(1,2,3)$
Let $C\subset\mathbb{P}^2$ be a plane quintic with an ordinary triple point in $[1:0:0]$, and two ordinary double points in $[0:1:0],[0:0:1]$, and otherwise general.
Then $C$ has geometric genus $1$. ...
- 281
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Conics and Hermitian curves over $\mathbb{F}_{q^2}$
Let $\mathcal{H} \subset \mathbb{P}^2(\mathbb{F}_{q^2})$ be the Hermitian curve, defined (up to projective equivalence) by
$$
X^{q+1} + Y^{q+1} + Z^{q+1} = 0.
$$
Its automorphism group is $\mathrm{PGU}...
- 21
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403
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Modular Interpretation of Boundary Strata of $ \overline{\mathcal{M}}_{g,n}$
The stratification of $\overline{\mathcal{M}}_{g,n}$ by so called stable graphs is a classical topic about Moduli of curves. A stratum corresponds to a decorated graph $\Gamma$ and in the literature, ...
- 203
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A morphism between the Teichmüller spaces
Let $T_{g,b}$ be the Teichmüller space of $\mathbb{C}$-curves $\Sigma_{g,b}$ with genus $g$ with $b$ marked points. The end of Section 4 of Drinfeld's famous "On Quasitriangular Quasi-Hopf ...
- 967
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Product of curves without negative correspondences
I suspect that the answer to this question is well-known to the experts. However, I was unable to find a reference. All answers or pointers to the literature will be appreciated.
Let $C_1$, $C_2$ be ...
- 68.7k
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Intersection of curves after a blow-up
Let $C,D\subset \mathbb{P}^n$ be irreducible and reduced curves, for some $n\ge 2$. For a point $p\in \mathbb{P}^n$ we can define the intersection multiplicity
$$I_{p}(C,D)=\mathrm{dim}(\mathcal{O}_{p,...
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Confusion on theorem in paper on monodromy of $p$-rank strata of moduli of curves
Application 5.7 in "Monodromy of the $p$-rank strata of the moduli space of curves", by Achter and Pries,
seems to imply that for $g \ge 3$, there exists a supersingular smooth projective ...
- 1,135
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1
answer
186
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Normality of the canonical dual of a very small line bundle
Let $C$ be a generic curve of genus $8$ and $D$ a $g^4_{11}$. Is $D$ $2$-normal? In other words, is the natural multiplication
$$H^0(D)\otimes H^0(D)\longrightarrow H^0(2D)$$
surjective?
More ...
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149
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The existence of special line bundles on algebraic curves
This is the problem that I encounter with when reading the proof of Theorem 17 of this paper.
Let $C$ be an algebraic curve of genus $g\geq3$. Assume that $L$ is a line bundle with $h^0(C,L)=3$. There ...
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How to find the minimal non-exact element for a polynomial curve
I have a question that I hope has a clear answer. I will begin by introducing some definitions and an example to help readers follow, assuming some background knowledge based on the tags I'm using.
In ...
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Curve of genus $g$ with more than $g-1$ automorphisms with disjoint graphs
This is a sequel to my previous question MO497996.
Let $\Sigma_g$ be a smooth, complex, projective curve of genus $g$. Then, the answer by Will Sawin to the question above shows that there are at most ...
- 68.7k
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267
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Maximum number of automorphisms with disjoint graphs on a projective curve of genus $g$
Let $\Sigma_g$ be a smooth complex projective curve of genus $g\geq 2$. Then, by the celebrated Hurwitz's automorphisms theorem, $\Sigma_g$ has at most $84(g-1)$ automorphisms, and this bound is sharp....
- 68.7k
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585
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How to find a suitable Input point for Satoh’s Miller’s inversion algorithms when subfield point compression is used with bn curves?
First, remember bn curves is a class of elliptic curves defined over curve $y^2=x^3+3$ with embedding degree 12 and $\mathbb G_2$ points lying over the curve twist $\frac {Y^2 = X^3 + 3}{i+9}$ defined ...
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411
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Maps from a hyperelliptic curve to $\mathbb{P}^1$
Given a smooth proper hyperelliptic curve $X$ over a field with a finite surjective map $f:X\to\mathbb{P}^1$, when does $f$ factor through the degree 2 covering $\pi:X\to\mathbb{P}^1$?
More precisely, ...
- 1,024
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A question about H. Darmon and A. Granville's paper "On the equations $z^m = F(x, y)$ and $A x^p + B y^q = C z^r$"
I want to ask about the reason of a conclusion in Section 4.1 of the following paper of H. Darmon and A. Granville:
On the equations $z^m = F(x, y)$ and $A x^p + B y^q = C z^r$.
Bull. London Math. Soc....
- 453
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580
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Plane sextics admitting a conic with 6 double-contact points
Let $S$ be a degree 6 (irreducible) complex plane curve. What is known about existence of a conic $C$ (a smooth degree 2 curve) touching $S$ in 6 double-contact points? I suspect that generically such ...
- 14.7k
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Transforming general square root equations into polynomial equations
Recently, I have encountered the following type equation
$$
\frac{f_{1}(x)}{g_{1}(x)\sqrt{h_{1}(x)}}+\frac{f_{2}(x)}{g_{2}(x)\sqrt{h_{2}(x)}}+\dotsb\frac{f_{n}(x)}{g_{n}(x)\sqrt{h_{n}(x)}}=\phi(x)
$$
...
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Does every (smooth) affine curve have a trivial canonical class?
Simple examples like quadrics, elliptic curves fits. Since for affine varieties we have non-constant regular function on it, it implies that canonical divisor is effective: f - regular, then df/dt is ...
- 121
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109
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Weil pairing of hyperelliptic curves over positive characteristics
Suppose $y^2=f(x)$ is a smooth projective hyperelliptic curve of genus $g$ over $\overline{\mathbb{F}_p}, p>2$. Its two torsion points on Jacobian can be represented as $c_i-c_j$ where $c_i$ ...
- 1,024
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Decomposition of spectrum of polynomial ring always possible?
Let $k$ be a field, $R = k[t]$ the coordinate ring in finitely many variables and $\text{Spec}(R)$ the spectrum of $R$. Is it always possible to write $\text{Spec}(R)$ as a union $\text{Spec}(R) = \...
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252
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Dimension of projective space in which moduli space of curves embeds
For some $g \geq 2$, let $\mathcal{M}_g$ be the coarse moduli space of smooth genus $g$ complex curves. There are a number of ways to see that $\mathcal{M}_g$ is a quasiprojective variety. What is ...
- 47.8k
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Is it impossible to extend powers of $\Lambda_{g-1}$ from $\Delta_0(M_g)$ to the moduli of curves $M_g$?
Consider the moduli of genus $g$ stable curves. Let $[C/p\sim q]$ be a moduli point of a curve in nodal curves divisor $\Delta_0$, then there is a natural map from degree $d$ polinomials on $H^0(\...
- 4,624
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Applications of the sheaf of invincible parts for (not families of) Gorenstein curves vs. jet bundles
I have some questions on the applicability of sheaves of invincible parts for, say, Gorenstein curves $C\to k$. The sheaves of principal parts are defined as double duals of the sheaves of principal ...
- 121
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Isotropic subgroup of Galois cover
Let $Y\rightarrow X$ be a ramified Galois cover between smooth algebraic curves. For a ramification point $y\in Y$, it is known that the isotropic subgroup $G_y$ is cyclic.
I am looking for a ...
- 1,969
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differentials on curves and its Jacobians
Let $(C, P)$ be a pointed curve over a field $k$ ($P$ is $k$-rational), let $J$ be the Jacobian of $C/k$. Then we can consider the canonical embedding $i \colon C \to J$ ; $Q \mapsto [\mathcal{O}(Q - ...
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Effective (algorithmic) computation of the moduli space of algebraic curves of genus $g$
Much literature (see, e.g., references here) has been written about the existence and “construction” of moduli spaces of algebraic spaces of a given genus — coarse or fine, smooth or stable, possibly ...
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Singularities of Algebraic Curves
As far as I know, the singularities of plane curves with degree $\leq7$ can be classified, see Chapter 2 of Makkoto Namba's text book 'Geometry of Projective Algebraic Curves'.
Also there are ...
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Equidimensionality of preimages under pure dimensional morphisms
Let $f:U \to W$ be a proper surjective map between locally Noetherian schemes of pure relative dimension $d$ (ie, every component of every fibre is of dimension $d$), such that $W$ is integral of ...
- 3,878
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Néron–Severi group of the product of two $G$-coverings
This is a follow-up to my previous question MO487259, so let me recall the set-up.
Let $G$ be a finite group and let $$\sigma_1=(g_1, \ldots, g_r), \quad \sigma_2=(h_1, \ldots, h_s)$$ be a $r$-tuple ...
- 68.7k
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Irreducible components of the fiber product of two $G$-coverings of $\mathbb{P}^1$
Let $G$ be a finite group and let $$\sigma_1=(g_1, \ldots, g_r), \quad \sigma_2=(h_1, \ldots, h_s)$$ be a $r$-tuple and a $s$-ple of elements of $G$, such that both of them generate $G$ and $$g_1g_2 \...
- 68.7k
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Symmetric product of $\mathbb{P}^{1}$
Let $S_{n}$ be the $n$-th symmetric group. I want to know why the symmetric product
$$(\mathbb{P}^{1})^{(n)}=(\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1})/S_{n}$$
is biholomorphic to $\mathbb{P}^{n}...
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Singularity of a rational plane cubic
(I asked the same question on Math Stack Exchange, but I received no answer.)
Let $C$ be a singular and irreducible plane cubic curve with integral coefficients. Is the singularity necessarily a ...
- 1,282
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189
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What is known about sub-bundles of the Hodge bundles over the moduli of curves?
E.g. what is known about the Harder-Narasimhan filtration of the Hodge bundle over this moduli ?
- 4,624
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79
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Multiplication by two in generalized Jacobian
Suppose that $X$ is a reduced proper algebraic curve over a finite field $k, char\neq 2$ with multiplicative singularities, namely the kernel $K$ of the map $Pic^0(X)\to Pic^0(\tilde{X})$ induced by ...
- 1,024
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How is the Bolza curve related to this quaternion algebra?
In the article Bolza surface, Wikipedia alludes to a connection between the Bolza curve (the complex algebraic curve of genus 2 with the largest symmetry group) and a certain quaternion algebra. But ...
- 24.7k
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267
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Are singularities of semistable models nice?
Let $X$ be a semistable curve defined over a number field $K$. Moreover, let $\mathcal X\to S=\operatorname{Spec }O_K$ be a (normal) semistable model of $X$. In this setting I would like to understand ...
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287
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Which étale double covers of plane curves come from conic bundles?
If we start with a smooth projective conic bundle threefold $p: X \to \mathbb{P}^2$, assuming that the discriminant $\Delta \subset \mathbb{P}^2$ is smooth, then we obtain a natural étale double cover ...
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139
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Are Hurwitz curves defined over real numbers / totally real fields?
A Hurwitz curve $C$ is a smooth projective complex curve (Riemann Surface) of genus $g \geq 2$ with precisely $84(g − 1)$ automorphisms.
By the proof of Hurwitz's theorem, there are finitely many $C$ ...
- 7,452
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1
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664
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Introductory references on curves over finite fields
In many papers on the function fields, it is commonly assumed that a curve is smooth, projective, and geometrically connected over a finite field. Could you please recommend the easiest (shortest) ...
- 1,143
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143
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Modular uniformization of curves over $\mathbb Q^{\mathrm{alg}}$ and abelian varieties of $\mathrm{GL}_2$ type over $\mathbb Q$
$\DeclareMathOperator\Jac{Jac}\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\GL{GL}\newcommand{\alg}{\mathrm{alg}}$Let $C$ be a (connected) smooth projective curve over $\mathbb Q^{\alg}$.
Let $X_1(...
- 7,452
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Belyi maps and defining Shimura curves over number fields
Let $(G, X)$ be a Shimura datum (so $G$ is a reductive group over $\mathbb Q$). For neat compact open $K \leq G(\mathbb A_f)$, we consider associated Shimura variety $M=Sh_K(G,X)$ with
$M(\mathbb C)= ...
- 7,452
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0
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152
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Computing with the Picard group of non-integral curves
Let $C$ be a curve over $\mathbb{Q}_p$ and let $\mathcal{C}$ be a regular model of $C$ over $\mathbb{Z}_p$, with $\mathcal{J}$ the Neron model of the Jacobian of $C$. Raynaud's theorem asserts that $\...
- 151
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192
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Universal semistable curve
For the moduli of stable curve, we have the result by Deligne-Mumford-Knudsen that there's an isomorphism of moduli spaces
$$\Phi : \overline{\mathcal{X}}_{g,n} \xrightarrow{\sim} \overline{\mathcal{M}...
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