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1
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1answer
94 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 ...
0
votes
0answers
45 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 ...
1
vote
1answer
96 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 ...
4
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0answers
72 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 ...
11
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0answers
164 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 ...
5
votes
0answers
115 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 ...
1
vote
1answer
161 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 ...
14
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0answers
585 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 ...
12
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0answers
458 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 ...
1
vote
0answers
114 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 ...
0
votes
1answer
95 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 ...
1
vote
1answer
74 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 ...
6
votes
1answer
331 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 ...
6
votes
2answers
275 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 ...
2
votes
0answers
119 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 ...
2
votes
1answer
65 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 $>1$ ...
3
votes
1answer
212 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 ...
9
votes
2answers
418 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 ...
2
votes
1answer
125 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. ...
2
votes
0answers
66 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 ...
2
votes
0answers
79 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 ...
6
votes
2answers
269 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, $|X(đť”˝_q)|=P(q)$. If in ...
2
votes
0answers
186 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 $[f:C\to X]$ ...
7
votes
2answers
362 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 ...
1
vote
1answer
119 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 ...
7
votes
0answers
269 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) = ...
5
votes
1answer
166 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 ...
0
votes
0answers
55 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 ...
1
vote
1answer
161 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 ...
2
votes
1answer
201 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 ...
2
votes
2answers
236 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 ...
1
vote
2answers
202 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 ...
2
votes
1answer
293 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 ...
0
votes
1answer
143 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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votes
0answers
139 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 ...
1
vote
1answer
167 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 ...
4
votes
1answer
108 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. ...
7
votes
1answer
200 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 ...
1
vote
0answers
87 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 ...
9
votes
1answer
250 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 ...
5
votes
1answer
171 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 ...
0
votes
0answers
79 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. ...
1
vote
1answer
48 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 ...
2
votes
2answers
164 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 ...
1
vote
2answers
185 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) = ...
2
votes
1answer
103 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?
1
vote
1answer
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 ...
4
votes
2answers
316 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 ...
2
votes
0answers
139 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 ...
9
votes
1answer
206 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 ...