# Tagged Questions

An algebraic curve or plane algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

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### About the difference variety of a curve

What is known about the singularity of the difference variety of a smooth curve $C$ of genus g>2? The difference variety is $C-C :=\{\mathcal{O}(p-q)| p,q\in C\} \subset Pic^0C$.
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### Where does the name $NE(X)$ come from?

Why do we call the cone of curves(effective one cycles) on a variety $X$ as $NE(X)$, what does $NE$ stand for?
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### Twisting locally free sheaves in characteristic $p$

Let $X$ be an irreducible nodal projective curve over an algebraically closed field of characteristic $p>0$. Denote by $\pi:\tilde{X} \to X$ the normalization of $X$. Recall, the short exact ...
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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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### 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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### 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. ...
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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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### 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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### 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 0\...
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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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### 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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### 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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### 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 ...