The algebraic-curves tag has no wiki summary.

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### 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 ...

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235 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 ...

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**1**answer

586 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, ...

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**1**answer

100 views

### Global sections in negative degree line bundles over singular curves

Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth ...

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201 views

### Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)

Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...

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**1**answer

197 views

### First chern class of fibers of compact Kaehler algebraic variety

Let $M$ be an compact Kähler algebraic variety and suppose $K_M$ is semi-ample. Consider the holomorphic map $\pi:X\to \Sigma \subset \mathbb CP^N$ with $Kod(M)=dim_\mathbb C\Sigma$ (here $Kod$ means ...

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101 views

### Families of curves with “almost-general” moduli

The Brill-Noether theorem says that, if $\rho(d, g, r) := (r + 1)d - rg - r(r + 1) \geq 0$, then there exists a unique component of the Hilbert scheme of curves of degree $d$ and genus $g$ in ...

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100 views

### rational sections of logarithmic differentials on a curve

Let $C$ be a smooth projective curve over a field $k$ of characteristic zero and $S$ a reduced divisor on $C$ (so just a collection of points). Consider the sheaf of logarithmic differentials ...

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78 views

### The minimum genus of a family of degree $12$ algebraic curves which comes from the resultant of two quartic polynomials

Let $f(t)$ be a rational normal cubic curve in $\mathbb{P}^3$ (it is not contained in any plane) and also we assume that this cubic curve passes through two points $(0,0,0)$ and $(1,0,0)$. By an easy ...

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77 views

### Obstruction to Gorenstein Liaisons of space curves

Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and ...

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824 views

### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...

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**1**answer

116 views

### Perfectness of the Jacobian of a curve

Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer ...

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74 views

### Do principally polarized abelian varieties enjoy a genus expansion?

This is a vague question from an interested outsider:
It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact ...

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133 views

### Neron model: can number of components decrease after based change?

Suppose I have Neron model over some discrete valuation ring.
Is there a result such that the number of components of the fiber over the closed point cannot decrease after some based change?
In ...

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90 views

### minimal conductors among elliptic curves with a fixed CM type

Let $K$ be a quadratic imaginary field. To simplify my life, let us assume
that $K$ has class number one.
Consider the following infinite set:
$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...

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199 views

### What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that ...

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138 views

### Strengthening of Suslin's rigidity argument?

To fix the situation, let $k$ be an algebraically closed field, and let $C$ be a smooth projective curve over $k$. Suslin's rigidity argument implies in particular that any class in ...

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**1**answer

197 views

### Moduli of curves in characteristic zero

Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}_{g,n}(K)$ and $\overline{M}_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing ...

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649 views

### New(?) reciprocity law

Consider three functions $f, g$ and $h$ on a smooth curve $X$ over $\mathbb{C}$.
I have found the following equality:
$$\sum (res(f\frac{dg}{g})\frac{dh}{h}-res(f\frac{dh}{h})\frac{dg}{g})=0.$$
Here ...

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507 views

### Degrees of maps from curves to $\mathbb P^1$

Let $a$ and $b$ be two relatively prime natural numbers. What is the largest number $c$ such that there is a curve with maps to $\mathbb P^1$ of degree $a$ and $b$ but no map to $\mathbb P^1$ of ...

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132 views

### Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...

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**1**answer

116 views

### Uniform position property and general hyperplanes

Given an irreducible curve $C$ of degree $d$ in $\mathbb{P}^r$ and a general hyperplane $H\subset\mathbb{P}^r$, the uniform position theorem states that any $r$ points on the hyperplane section $H\cap ...

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88 views

### Invertible functions on open subset of hyperelliptic curve

Let $C \to \mathbf P^1$ be a hyperelliptic curve of genus $g \ge 2$ obtained as a double cover of $\mathbf P^1$ branched at $r$ points. Let $\tilde U\subset C$ be its open subset obtained by removing ...

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376 views

### Understanding of Tamagawa numbers of hyperelliptic curve

One's can find following definition of tamagawa numbers in Dino Lorenzini paper "Torsion and Tamagawa numbers":
Let $K$ be any discrete valuation field with ring of integers $O_K$ ,
uniformizer ...

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**2**answers

404 views

### Calculate reduction of Jacobian of hyperelliptic curve

Suppose I have a hyperelliptic curve of genus $2$ over $\mathbb Q$. I want to get information about its Jacobian reduction at prime $p$ (especially, in case $p=2$). Also I'm interesting in the group ...

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123 views

### A generalization of the Weil reciprocity law in a case of any two sections of line bundles on a curve

It seems to me that there should exist a generalization of the Weil reciprocity law on curves, where instead of functions one takes arbitrary sections of two line bundles.
More precisely, it might ...

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**1**answer

105 views

### Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication

Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ ...

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261 views

### Conditions for a parametric curve to avoid self-intersection?

Suppose a planar curve $C$ is defined by parametric
equations in $t$: $x(t)$ and $y(t)$.
Are there conditions on these two functions that guarantee
that $C$ does not self-intersect?
For example,
the ...

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556 views

### Rigorous version of heuristic argument for genus-degree formula?

A recent MO question about non-rigorous reasoning reminded me of something I've wondered about for some time.
The genus–degree formula says that genus $g$ of a nonsingular projective plane curve of ...

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141 views

### A problem related to deformation of irrational curves

The following question arises from the proof of "bend-and-break" lemma:
Let $X$ be a projective variety over $\mathbb{C}$ and $C$ be an irrational smooth curve. Let $c \in C$ be a fixed closed point. ...

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71 views

### A basic question on complete intersection liaisons of curves

I am a beginner in the Linkage theory and would like to clarify certain points I am not sure of.
Let $P$ be the Hilbert polynomial of a curve in $\mathbb{P}^3$. Let $L$ be an irreducible component of ...

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### Moduli space of sheaves on a ribbon

In the paper "A non-linear deformation of the Hitchin dinamycal system", Donagi-Ein-Lazarsfeld describe the irreducible components of the moduli space $\mathcal M_R$ of stable sheaves of numerical ...

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285 views

### Varieties with polynomial point count and no odd cohomology

Let $X$ be a proper algebraic variety. $X$ is said to have polynomial point count if there is a polynomial $P$ such that for all finite fields $\mathbb F_q$ with $q$ elements, $|X(𝔽_q)|=P(q)$.
If in ...

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233 views

### Beautiful curves in Gromov-Witten theory, and in Donaldson-Thomas theory

Let $X$ be a smooth complex projective threefold, and $\beta\in H_2(X,\mathbb Z)$ a curve class. In the Kontsevich's moduli space of stable maps, $\overline M_g(X;\beta)$, a general point $[f:C\to X]$ ...

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401 views

### Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is

Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.
If $\mathcal X$ can be defined over ...

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144 views

### Unique decomposition of locally free sheaf

Below let's work over coherent sheaves on a smooth projective algebraic curve.
We call a subsheaf $\mathcal{F'}$ of $\mathcal{F}$ saturated if it $\mathcal{F/F'}$ is locally free.
We call a locally ...

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313 views

### The curve $(x+y+z)^3=27xyz$

Can someone point me to literature about the curve defined by $F(x,y,z):=(x+y+z)^3-27xyz$? I'm sure this curve must be well-studied, due to the remarkable property that
$$
F(x^3,y^3,z^3) = ...

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179 views

### Units of $\mathbf Z[X,Y]/(P(X,Y))$

Let $P(X,Y)\in \mathbf Z[X,Y]$ be an irreducible polynomial and let $A$ denote the quotient ring $\mathbf Z[X,Y]/(P)$.
What is known about the group of units of $A$?
It's not even clear to me that ...

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58 views

### plotting parametrized algebraic curves near singularities

I have a parametrized algebraic curve:
x(t)=A(t)/D(t);
y(t)=B(t)/D(t);
with A(t) and B(t) being polynomials in t. The curve is solution of a linear system in two unknowns x and y with Cramer's ...

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180 views

### Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem:
If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most ...

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217 views

### Surfaces singular along a curve

Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...

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280 views

### Etale covers of a hyperelliptic curve

Let $X$ be a hyperelliptic curve of genus at least two.
Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve.
Is $Y$ hyperelliptic?
More ...

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242 views

### multi-tangent space for algebraic curves

For a general plane curve of degree $\ge 4$, the number of bitangent lines is known. Also, I found that the number of tritangent planes have been worked out for some space curves given by intersection ...

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308 views

### degree 7 rational curves through ten points in P4

This is a very classical flavoured question, and probabaly it is not difficult. I would like to know the shape of the space of rational degree 7 curves in $P^4$ that pass through 10 fixed points. By ...

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151 views

### Curve of degree $d$ through $2d+1$ points in $\mathbb P^3$

It is known that a Hilbert scheme of degree $d$ curves in $\mathbb P^3$ can have dimension more than $4d$. But, does it imply that for some types of curves there are such a curve through any, say, ...

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### Totally real points on curves

Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...

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180 views

### Counting curves of degree 4 in $\mathbb{P}^{3}$

Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...

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### Normalization of a curve and push forward of vector bundles

Let $C$ be a projective curve (over an algebraically closed field, not necessarily of characteristic zero) which is smooth except for exact one node. Let $\pi:\tilde{C} \to C$ be its normalization. ...

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248 views

### Finite morphisms to projective space

Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.
Let $d^\prime ...

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97 views

### Degeneracy divisor of the “trace” morphism

Let $f\colon X\to Y$ be a finite morphism of smooth curves over an alg. closed field of characteristic zero. I recently asked how methods reminiscent of basic algebraic number theory can be used to ...