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5
votes
1answer
179 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 ...
1
vote
1answer
210 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 ...
1
vote
0answers
188 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 ...
2
votes
1answer
270 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 ...
1
vote
1answer
124 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 $C$, ...
4
votes
0answers
210 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 ...
7
votes
1answer
182 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 ...
6
votes
1answer
359 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 ...
1
vote
1answer
209 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 ...
0
votes
1answer
259 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? ...
2
votes
0answers
113 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 $C_{red} \hookrightarrow \mathbb{P}^n$ ...
1
vote
0answers
100 views

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 ...
1
vote
1answer
123 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 ...
7
votes
1answer
316 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 ...
3
votes
1answer
121 views

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) = ...
0
votes
1answer
379 views

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 ...
4
votes
2answers
202 views

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 $m$, ...
1
vote
2answers
338 views

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 ...
2
votes
2answers
270 views

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 ...
2
votes
0answers
104 views

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-$p$ ...
11
votes
2answers
861 views

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 ...
3
votes
1answer
184 views

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 ...
0
votes
2answers
302 views

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 ...
1
vote
0answers
142 views

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 ...
1
vote
1answer
210 views

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 $\mathbb R$, ...
2
votes
0answers
161 views

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 ...
0
votes
1answer
222 views

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.
1
vote
1answer
188 views

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 ...
1
vote
1answer
139 views

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 $\mathbb{E}$ ...
6
votes
1answer
425 views

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 ...
0
votes
1answer
141 views

Totally tangent planes to a curve in $\mathbb{P}^3$

How many planes are totally tangent to a curve of $\mathbb{P}^3$ which is intersection of a generic quadric and a generic cubic? Or equivalently, considering the cubic as a blow up of $\mathbb{P}^2$ ...
1
vote
1answer
209 views

Automorphisms of rational (connected) projective curves

To fix the ideas all curves are supposed to be defined over $\mathbb{C}$. Let $C$ be a rational connected projective curve. Note that we don't assume the curve to be smooth. Let $Aut(C)$ be the group ...
2
votes
1answer
181 views

F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$

It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all ...
0
votes
0answers
117 views

Existence of a curve with no points over finite separable field extensions

Does there exist a field $K$, and a smooth projective geometrically connected curve $C$ over $K$ such that, for all finite separable field extensions $L/K$ the curve $C$ has no $L$-rational points? I ...
0
votes
0answers
51 views

open subset in constructible set of divisors

Let a smooth projective curve $X$ over $\mathbb{C}$. Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$. Let $N=\deg (D)$ and ...
6
votes
0answers
513 views

Bezout Theorem in $\mathbb P^3$

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 $d$ curve passes then $d^2\geq ...
1
vote
0answers
61 views

lift sections on a thickened curve

Let $X$ a curve over an algebraically closed field $k$ and $D$ a divisor on X. Fix an integer $N$ and a closed point $x$ on $X$, we assume that $\deg(D)$ is big enough such that we have a surjective ...
6
votes
0answers
180 views

Counting plane curves over various fields

Fix two integers $d$ and $g$. The number of genus $g$ and degree $d$ curves passing through $3d+g-1$ generic points on the complex projective plane is finite and doesn't depend on the choice of ...
9
votes
2answers
215 views

Effectiveness of the distinguished theta characteristic in characteristic 2

Let $k$ be an algebraically closed field of characteristic 2. Let $C$ be a (smooth projective connected) curve over $k$. Can there exist a rational function on $C$ whose differential is holomorphic ...
3
votes
3answers
361 views

General curves of genus 3 as plane sections of Kummer surfaces

Is it true that a general curve of genus 3 is a plane section of an appropriate Kummer surface in $\mathbb P^3$? By Kummer surface I mean image of a principally polarized Abelian surface w.r.t. the ...
2
votes
2answers
247 views

Determine asymptotic behavior of algebraic curves

Take an example polynomial $f(x, y) = y^2 x + y^3 - x^2$. A solution to $f(x,y)=0$ exists with Puiseux series given by $y(x) = x^{2/3} - x/3 + x^{4/3}/9+\cdots$. I got this by having Mathematica ...
1
vote
2answers
400 views

Equations of elliptic curves

First part of question I have asked on mathoverflow already: http://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve 1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...
3
votes
3answers
312 views

Reference for hyperelliptic curves

I was reading a paper the other day that said that all automorphisms of a hyperelliptic curve are liftings of automorphisms of $\mathbb{P}^1$ operating on the set of branch points. Can someone point ...
5
votes
2answers
1k views

What does the Jacobian of a curve tell us about the curve?

A natural object in the study of curves is the Jacobian of a curve. What are some natural geometric properties of the curve that the Jacobian encapsulates? In other words, what can the Jacobian tell ...
-1
votes
1answer
139 views

effective divisors on a curve and upper semi-continuity

Let consider a smooth projective curve $X$ over $\mathbb{C}$. We consider the scheme that classifies effective divisors of degree $d$, which is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the ...
1
vote
1answer
228 views

a question on the space of divisors on a curve

Let $X$ a complex curve and $x\in X$ a point. We consider the space of effective divisors $D$ with fixed degree $d$, whic we know is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the symmetric group. ...
8
votes
3answers
744 views

What prevents a cover to be Galois?

Let $f:X\rightarrow Y$ be a ramified cover of Riemann surfaces or algebraic curves over $\mathbb{C}$. My question is can one in terms of the ramification data of $f$, determine whether the cover is ...
13
votes
1answer
305 views

Curves which do not dominate other curves

Let $g>1$ be an integer. Does there exist a (smooth projective) genus g curve $X$ which doesn`t dominate a curve of positive genus and genus smaller than $g$? Surely such curves exist. Just take a ...
4
votes
2answers
140 views

Dimension of the space of invariant quadratic differentials in Galois covers

Let $f: X \rightarrow Y $ be a Galois cover of with $X$ and $Y$ algebraic curves over $\mathbb{C}$. I want to compute the dimension of the subspace of $G$-invariants in $H^{0}(X,\omega^{\otimes2})$ ...
5
votes
1answer
499 views

Analogy between Jacobian of curve and Ideal class group

It is excerpt from "Algebraic Geometry Codes Basic ...