The algebraic-curves tag has no usage guidance.

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### Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...

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158 views

### Existense of semi-stable vector bundles on smooth curves in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic and $X$ be a smooth projective curve over $k$ of genus $g \ge 2$. Fix a polarization $L$ on $X$. Does there exist a semi-stable ...

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### Gonality and Clifford dimension of curves on a K3 surface

Let $X$ be a K3 surface. Let $L$ be an ample line bundle on $X$. When/how can we say that any smooth curve $C\in |L|$ has maximal gonality $k=[\frac{g+3}{2}]$ and Clifford dimension 1. Is there some ...

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164 views

### Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.
Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...

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### cross sections of semialgebric sets

Let n be bigger than two, and let A be a subset of the n-dimensional euclidean space. Suppose that the intersection of A with any (n-1)-dimensional affine hyperplane is semialgebraic. Can one conclude ...

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### Standard techniques on rationally connected varieties

Is there some standard technique or approach to determine when a (irreducible) subvariety of a rationally connected variety is again rationally connected? Any reference/text dealing with this kind of ...

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122 views

### vanishing locus of a certain section of $\wedge^{2n} (\mathcal{O}^{2n}_{M \times M})$

(sorry, if this question looks a bit arbitrary; I tried to think about it and couldn't make much progress; so even a suggestion on how to approach this would be very welcome)
Let $M$ be a curve over ...

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137 views

### A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem

I propose a conjecture variant of Cayley-Bacharach's theorem.
I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...

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171 views

### A conjecture associated with n-gon cut curve of degree m

This conjecture is a generalization of A theorem for cubic-A generalization of Carnot theorem, in MSE question. I'm an electrical engineer, I am not a mathematician. I don't know how to prove this ...

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### A question about complex plane algebraic curves

I would like to ask a question about plane projective curves. Let $C\subset{\mathbb P}_2={\mathbb P}(V)$ be a plane curve of degree $n\geq 3$. Then we have a non splitted exact sequence
...

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### The line bundle of the divisor at infinity of the moduli stack of stable curves of genus $g \ge 2$

Let $\overline{\mathscr{M}}_g$ be the $\mathbb{Z}$-algebraic stack of stable curves of genus $g \ge 2$, as constructed in the paper of Deligne and Mumford. The degeneracy locus of the universal stable ...

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76 views

### A property of the semi-local ring of the normalization of a singular curve

I have two following questions.
1) Let $R$ be a local ring in an algebraic function field of one variable over an algebraic closed field $k$. Let $\bar{R}$ and $m$ be its integral closure and maximal ...

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### Families of trigonal curves with hyperelliptic limit

Suppose I have a family of trigonal curves $C\to D$ over a closed disk $D$ where the central fiber $C_0$ is hyperelliptic (this is of course possible since the hyperelliptic locus is in the closure of ...

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### Motivic fundamental group of the moduli space of curves?

Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...

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64 views

### Ring of functions regular away from $\infty$ of an algebraic curve

Suppose $X$ is a smooth, geometrically irreducible projective algebraic curve over the finite field $\mathbb{F}_q$ and fix a closed point $\infty \in X$. Denote by $A = \Gamma(X - \{\infty\}, ...

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### Intuition behind if neither $D$ nor $K-D$ are equivalent to an effective divisor, then $\deg(D) = g-1$?

Is there any intuition behind the following fact?
If neither $D$ nor $K-D$ are equivalent to an effective divisor, then $\deg(D) = g-1$.
Here, $K$ is the canonical divisor. It means the degrees ...

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### Set of smooth curves on the Hilbert scheme is open. H

Let $H = Hilb_{d,g,r}$ be the Hilbert scheme of genus $g$ curves of degree $d$ in proyective space $\mathbb{P}^r$, over an algebraically closed field $k$.
Is it true that the set of points of $H$ ...

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115 views

### Existence of a map between curves

Given two algebraic curves defined over the rationals, is there a method for determining whether there exists a surjective map from one curve to the other? For instance, suppose X and Y are affine ...

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168 views

### Bézout's theorem for arcs in the plane

Consider two polynomials $p,q \in {\mathbb R}[x,y]$, both of degree $d$. Let $\gamma_p$ and $\gamma_q$ be the two curves in ${\mathbb R}^2$ that are defined by these polynomials, and assume that these ...

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### Is there a concrete description of $\Theta_{sing}$ for a generic curve of genus 6?

If C is generic of genus 6, then $\Theta_{sing}$ is a smooth surface. Can anyone give me a reference or a hint as to what that surface might be. What are the numerical characteristics of this surface? ...

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### Non canonical singularities of moduli spaces of curves

Is it true that for any $g\geq 1$ and $n$ such that $\overline{M}_{g,n}$ has dimension at least two the locus in $\overline{M}_{g,n}$ parametrizing reducible curves which are union of an elliptic ...

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### Riemann-Roch formula for nodal curves

Let $X$ be an irreducible, reduced, projective curve over an algebraically closed field, with at worst nodes as singularities. Let $\mathcal{F}$ be a trivial vector bundle on $X$ of rank $r$. Consider ...

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### Model over DVR for smooth projective curves

Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...

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158 views

### Different proof's of Marten's theorem

I am referring to Marten's theorem on the dimension of $W_d^r $ as in ACGH p. 192 . It seems to me that an even shorter proof can be given using Hopf's Theorem that if $\nu : A \otimes B \to C $ is ...

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239 views

### Theta characteristics of genus$\geq3$ curve

Let $C$ be a smooth curve of genus$\geq3$ over $\mathbb{C}$, so there are $2^{g-1}(2^g-1)$ odd theta characteristics and $2^{g-1}(2^g+1)$ even theta characteristics. Do we know how many of them has ...

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### Morphisms contracting a family of curves

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve.
...

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### The significance of the Parvaresh-Vardy curve

Even though this seems like a computer science question, it is purely mathematical and concerns polynomials and curves over finite fields.
Consider the Parvaresh-Vardy list decoder.
As I understand ...

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### Does this extension of Hodge structures split over $\mathbb{Q}$?

Let $X$ be a smooth projective curve of genus $\geq 1$ over $\mathbb{C}$, $H^\cdot=H^\cdot(X)$, and $K$ be the kernel of cup product $\cup: H^1\otimes H^1\rightarrow H^2$. Consider the extension of ...

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### Can every genus $2$ curve be written as ramified cover of elliptic curve?

Suppose $C$ is a curve of genus $2$, does $C$ admit a surjective morphism onto some elliptic curve $E$?

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### Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?

See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background.
If ...

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### Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations?

Suppose $k$ is a field of characteristic zero, and $R$ is a $k$-algebra. The $R$-module of Kähler differentials $\Omega_{R/k}$ of $R$ over $k$ with generators $\{dr\}_{r \in R}$ is the module subject ...

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### Rational curves through a fixed number of points

Let us fix two positive integers $d$, and $N$. Can we determine a third integer $n$ such that given $n$ general points $p_1,...,p_n\in\mathbb{P}^N$ there exists a unique rational curve of degree $d$ ...

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### Rational curves in projective spaces

Let $X\subset(\mathbb{P}^{N})^n$ be the variety defined as follows: $(p_1,...,p_n)\in (\mathbb{P}^{N})^n$ such that there exists a rational curve $C$ of degree $d$ with $p_1,...,p_n\in C$.
Is there a ...

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### On the compactification of moduli space of vector bundles

Let $X$ be an irreducible, nodal curve over an algebraically closed field of genus at least $2$. Denote by $U(r,d)$ (resp. $U^0(r,d)$) the moduli space of torsion-free (resp. locally free) sheaves of ...

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### Does there exist a continuous surjection? [closed]

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...

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### Is every proper regular relative algebraic space curve over a Dedekind domain projective?

This question is in some sense a follow up to a related question Is a normal proper relative curve over a DVR projective?
Let $R$ be a Dedekind domain, let $S := \mathrm{Spec}(R)$, and let $X ...

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### Do discrete valuation rings correspond to local rings of points in fibre?

Given projective curves $C$ and $C'$ and a surjective morphism $\varphi\colon C\to C'$, such that $Q\in C'$ is a smooth point and its fibre $\varphi^{-1}(Q)$ consists of smooth points.
Then ...

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### Embedding curves in hypersurfaces

Consider a curve $C$ in $\mathbb{F}_q^m$, say. I am interested in the existence of curves not contained in any small degree hypersurface.
For instance, a helix is not contained (or non-embeddable) in ...

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### Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?

Let $\mathcal{M}_2$ be the moduli space of genus two curves and $\mathcal{A}_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding ...

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### Graded ring of a genus 2 curve

Let $X$ be a smooth projective complex curve of genus 2 with canonical divisor $K$. $X$ of course is hyperelliptic and has an involution that I denote by $j$.
There exists 3 possibilities for ...

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### Are two line bundles with the same ramification type necessarily isomorphic?

I have no motivation for the following problem, I am just curious if it is true or not. Here it is:
If $l_1$ and $l_2$ are two complete $g^r_d$'s on a smooth curve $C$ such that the vanishing ...

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### Example of a genus-1 degree-7 plane curve

I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1.
One can show that for a general ...

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### Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?

Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset ...

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### Elliptic Curve Multiplication [closed]

What would happen if I performed Elliptic Curve multiplication on some random point within the FiniteField that wasn't actually on the curve? I assume that I would get a point in return but would that ...

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### Is there any explicit result on the triangulated category of singularities of a curve?

This question is related to this MO question.
Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category ...

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### What is the relation between the $K_0$ of a singular curve and its normalization?

Let $X$ be a singular curve over a field $k$. We define $K_0(X)$ to be the Grothendieck group of the category of coherent sheaves on $X$.
For $X$ we have its normalization $\widetilde{X}$ and hence ...

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### Are curves over imperfect fields defined over a smaller field?

Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...

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### the associated action on the transition functions

Let $X$ be a curve with an involution $\sigma$ generically unramified, given a $G-$bundle $E$ of rank $r$, than we ca take its pull-back, I want to describe the action of $\sigma$ on $G$. Fix a ...

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### Construction of curves and morphisms

Given any triple of positive integers $(g',g,d)$ with $2g'-2\geq d(2g-2)$.
Does there always exist curves $C_{g'},C_g$ of genus $g',g$ with a degree $d$ morphism $f\colon C_{g'}\to C_g$?
If we fix ...

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### Reflexive sheaves on stable curves

Let $C$ be a stable curve over an algebraically closed field of positive characteristic and $\mathcal{F}$ be a reflexive sheaf on $C$. Is $\mathcal{F}$ locally free?
EDIT Is the projective dimension ...