The algebraic-curves tag has no wiki summary.

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### Non-abelian group from affine hermitian curve

I was playing with the Hermitian curve $y^q + y = x^{q+1}$ over the field $GF(q^2)$ and chanced upon the following (non Abelian) group law on the points of the affine curve:
$(a,b) * (c,d) = ...

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### Global sections of the structure sheaf of a non-reduced projective scheme

Let $C$ be a curve in $\mathbb{P}^3$ which is of local complete intersection is some smooth surface in $\mathbb{P}^3$. Assume $C$ is non-reduced. What can we say about $h^0(\mathcal{O}_C)$? When can ...

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### A question on the existence of the quotient of the Hilbert scheme of tricanonical curves

In order to construct the coarse moduli scheme of smooth projective curves of genus $g$, the classical results of Mumford (using the numerical criterion of stability) say that for large enough $m$, ...

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### Connections on the Hodge bundle?

Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of ...

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### spin bundle vs. hodge bundle

Let $\overline{\mathcal{M}}^r_{g,n}$ the space of $n$ pointed $r$-stable curves of genus $g$, endowed with a root of the canonical sheaf, and let $\mathcal{C} \to \overline{\mathcal{M}}^r_{g,n}$ be ...

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### Automorphisms of function fields under constant reduction

Let $K=\mathbb{Q}(x,y)$ be a function field of genus at least 2, with defining equation $f(x,y)=0$ (say, absolutely irreducible and with coefficients not divisible by $p$), and let $k$ be the mod-$p$ ...

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### Permuting collinear points on a curve

Let $C \subset {\bf CP}^2$ be an irreducible algebraic smooth (projectively) planar curve over the complex numbers of degree $d$ (we allow finitely many points to be deleted from $C$ to make it ...

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### Etale covers of products of curves

Is a finite etale cover of a product of curves again a product of curves?
The answer is no in general. Here's one way to construct an example. Take the product $A$ of two elliptic curves and an ...

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### Rationality of curve does not depend on base change

By a curve I mean an integral one-dimensional scheme of finite type over a spectrum of a field.
Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is ...

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### Hopf lemma for line bundles on curves in algebraic geometry

In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one ...

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### curves in varieties

Let $V$ be an affine algebraic variety defined over $\mathbb R$.
We assume that $V\subset \mathbb A_n$(affine $n$ space).
Suppose that for any algebraic curve $C$ in $V$ defined over $\mathbb R$, ...

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### branch locus of the discriminant map $\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of pointed, stable, genus $g$ curves. Let $\overline{\mathcal{H}}_{g',r}$ be the hurwitz space of cyclic covers of degree $r$ of genus $g$ curves ...

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### Finite number of points as intersection of two plane curves

How to prove that any finite set in the affine plane is realizable as intersection of TWO plane curves?
Thanks.

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### Complete Linear system on Del Pezzo surfaces

Is there always a reducible curve (EDIT: with exactly two irreducible components intersecting in at least 2 points) in a complete linear system (EDIT: of dimension at least 2 with curves of genus at ...

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### Hodge bundle on F-curves

Let $\mathbb{E}\rightarrow\overline{M}_{g,n}$ be the Hodge bundle. Let us cosider an $F$-curve of type $\overline{M}_{1,1}\subseteq\overline{M}_{g,n}$. Is the degree of the restriction of $\mathbb{E}$ ...

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### Nagata's conjecture in positive characteristic

For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ irreducible reduced curve passes then ...