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### 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 ...

**10**

votes

**2**answers

588 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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**1**answer

144 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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72 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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94 views

### 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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289 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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**0**answers

239 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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408 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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**1**answer

155 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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317 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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**1**answer

180 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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**1**answer

181 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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votes

**1**answer

220 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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**2**answers

294 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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vote

**2**answers

255 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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**1**answer

310 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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**1**answer

152 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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152 views

### 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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**1**answer

185 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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**1**answer

123 views

### 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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**1**answer

267 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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**0**answers

102 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 ...

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291 views

### Showing that $2c_1(f_*\mathscr O_X)=-f_*R_f$ on curves, maybe by local fields

I originally asked this question on Mathematic StackExchange, but it did not seem to be attracting any attention, so now I am trying mathoverflow. I hope it is not too simple or unappropriate a ...

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**1**answer

210 views

### general position lemma for tangent lines of an algebraic curve

Let $X$ be a smooth irreducible algebraic curve in $\mathbb{P} V$.
The general position lemma states that the points given by general hyperplane section of $X$ are "in general position". I'm ...

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89 views

### Embed the normalization of a curve in a larger space

I'd like to believe that this problem has a positive answer, but I don't know a nice reference. Actually I've never worked with embedded curves, so I apologize in advance if the question is too silly.
...

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**1**answer

50 views

### Function field Towers of larger depth of recursion

A function field tower is a sequence of function fields
$$\mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \dots \subset \mathcal{F}_{n} \subset \mathcal{F}_{n+1} \subset \dots $$
over a base ...

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votes

**2**answers

196 views

### Curve of 3-secant lines

Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.
Consider the variety ...

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**2**answers

197 views

### factorizing a quartic plane curve as $f_3f_1-f_2^2$

Let $C$ be a quartic plane curve. Suppose that for a given coordinate system
$C=(F_4(x_0,x_1,x_2)=0)$ where there the polynomial $F_4$ factorizes as
$$
F_4(x_0,x_1,x_2) = ...

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**1**answer

107 views

### When is the Clifford index of a curve computed by pencils?

Under which circumstances is the Clifford index of a curve computed by pencils?

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**1**answer

69 views

### Determining the desingularization from the complete local ring

Suppose I have a curve $C$ over a field $k$ and that $p$ is a singular point of $C$. Let $f : X \to C$ be the desingularization of $C$ at $p$. Then for each $s \in f^{-1}(p)$ we have a map of local ...

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**2**answers

349 views

### When are arithmetic and geometric monodromy equal?

Let $f: Y\to X$ be a finite separable morphism of curves over the finite field $\mathbb{F}_q$. Is there a simple condition under which the arithmetic and geometric monodromy of the covering are equal ...

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**0**answers

150 views

### When are Hilbert schemes connected by piecewise smooth curves?

Are there examples of Hilbert scheme $H$ of curves in $\mathbb{P}^3$ such that there exists an irreducible component $L$ of $H$ such that for any two points in $L$, there exist smooth projective ...

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**1**answer

227 views

### Minimal number of intersection of curves in $\mathbb P^2$

Let $C_1$ and $C_2$ be two smooth curves of degrees $m$ and $n$ in $\mathbb CP^2$. By Bezout's theorem the maximal number of their intersections is $mn$. I wonder if the minimal possible number is ...

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**1**answer

120 views

### Castelnuovo-Mumford regularity of curves

Let $C$ be a projective curve (scheme of pure dimension $1$). This induces a short exact sequence $$0 \to \mathcal{I}_C \to \mathcal{O}_{\mathbb{P}^n} \to i_*\mathcal{O}_C \to 0$$ for some $n$ such ...

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**0**answers

132 views

### Deformation of complete intersection curves

Let $\mathcal{X} \to B$ be a family of smooth surfaces in $\mathbb{P}^3$. Let $\mathcal{C} \to B$ be a family of curves in $\mathbb{P}^3$. Assume that for all $b \in B$, the fiber $\mathcal{C}_b$ is ...

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125 views

### On a difference between $i_!$ and $i_*$ over $\mathbb{P}^1$

Let $X$ be a smooth projective surface in $\mathbb{P}^3$ containing a line $l$.
Denote by $C$ the curve corresponding to the divisor $2l$. Let $p \in C$ be a closed point. Denote by $U:=C \backslash ...

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**1**answer

166 views

### Higher Weierstrass points on curves of genus 3

So this question is directly related to a comment made by David Mumford in his
Lecture 1 given at U. Michigan in 1974 entitled: What is a curve and how explicitly can we describe them ?
Mumford ...

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**1**answer

205 views

### Does the normalization morphism induce isomorphism on residue fields?

The question is basically coming from the following situation:
Let $C$ be an integral curve over a field $k$ (EDIT and assume that $k$ is not algebraically closed) and let $\phi\colon C^N\to C$ be the ...

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185 views

### Pull-back of globally generated sheaves

Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...

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**1**answer

263 views

### How to obtain the Period matrix from the Igusa Invariants of a genus two curve?

I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely:
I consider a family of genus two ...

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**1**answer

121 views

### Restriction of sheaves on curves

Let $C$ be a scheme of pure dimension $1$. Let $C_1$ be a closed subscheme of $C$ of pure dimension $1$. Denote by $i:C_1 \hookrightarrow C$ a closed immersion. Given a sheaf $\mathcal{F}$ on $C$, ...

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196 views

### Deligne-Mumford moduli spaces and compactification of symmetric matrices

The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered ...

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**1**answer

180 views

### Projective embedding in families of curves

Let $\pi:\mathcal{X} \to B$ be a family (flat, projective, surjective morphism) of projective curves (not necessarily reduced) where $B$ is smooth, irreducible. Suppose that for some closed point $b_0 ...

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**1**answer

354 views

### Can any curve be embedded into $\mathbb{P}^3$?

We know from Hartshorne's "Algebraic geometry" that if a curve $C$ is smooth then there exists a closed immersion of $C$ into $\mathbb{P}^3$. Does this still hold if $C$ is not generically reduced or ...

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203 views

### Weierstrass points on modular curves

What is knowns about Weierstrass points on modular curves? Are there any explicit formulas of them, or any information about Weierstrass gaps? I am interested in (compactifications of) the quotients ...

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246 views

### Example of non-vanishing of first cohomology of a torsion coherent sheaf on a curve

By a curve we mean a projective scheme of pure dimension one. Can some one give an example of a curve $C$ and a torsion coherent sheaf on $C$ such that its first cohomology group does not vanish?
...

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112 views

### Projective embedding of non-reduced curves

By curve I mean a scheme of pure dimension $1$. Let $(\Gamma(C_{red},\mathcal{L}))$ be a complete linear system on $C_{red}$ which gives a projective embedding $C_{red} \hookrightarrow \mathbb{P}^n$ ...

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### Surjectivity of global sections of sheaf of Kaehler differentials

By a curve I mean a scheme of pure dimension $1$.
Let $C_1, C_2$ be a local complete intersection curve in $\mathbb{P}^3$ such that $C_1 \cap C_2$ are finitely many points. Assume further that ...

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**1**answer

120 views

### divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves

Inside the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}_{0,4}$, and those of ...

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293 views

### Singularities of moduli spaces of curves

Let $\overline{M}_{g,n}$ be the moduli space of $n$-pointed genus $g$ Deligne-Mumford stable curves. This is a normal projective scheme. Then
$$codim_{\overline{M}_{g,n}}Sing(\overline{M}_{g,n})\geq ...