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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152 views

### For a curve $C$ and its Jacobian $J$, is $\Gamma(J, \Omega) \to \Gamma(C, \Omega)$ an isomorphism?

Let $k$ be a field, $C$ a smooth complete $k$- curve, $J$ its Jacobian variety, $P \in C(k)$ a rational point, $f : C \to J$ the morphism associated with $P$.
Then does $f$ induce an isomorphism $\...

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114 views

### Reference request: Vanishing of first cohomology term in Riemann-Roch theorem for singular projective curves over a field

$\newcommand{\F}{\mathcal{F}}$
$\newcommand{\ox}{\mathcal{O}_X}$
Let $f:X \to \operatorname{Spec}(k)$ be a projective scheme of dimension one over a field $k$. The Riemann-Roch equation for such ...

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89 views

### Principally Polarized CM Abelian Variety

I am interested in considering examples of abelian varieties that are principally polarized with CM in dimension three. However, I am struggling to construct or find even a single instance.
In ...

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**1**answer

131 views

### difference between Cohen Macaulay and locally Cohen Macaulay curve

I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves.
The stacks project https://stacks.math.columbia.edu/tag/02IN says that a scheme (a curve in ...

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**1**answer

416 views

### Are relative curves $X \to S$ determined by their fibers?

Consider relative curves $X \to S$, defined to be flat, integral, projective schemes of relative dimension 1 over $S$. When are these objects determined by their fibers?
So if $X,Y$ are $S$-schemes ...

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25 views

### Algebraic connection of the quadratic Lyapunov function existence with the Hurwitz criterion for third order linear system

In what publication the following theorem is proved?
Let $a$,$b$, $c$ be real numbers and we choose a Lyapunov function for the system:
$$
\left\{
\begin{array}{cr}
\dot{x} = y \\
\dot{y} = z \...

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88 views

### When does the Jacobian of a smooth curve contains an unique principal polarization

Let $X$ be a smooth, projective curve of genus at least $4$ and $X$ non-hyperelliptic. I am looking for additional conditions on $X$ such that the Jacobian $J(X)$ of $X$ contains an unique principal ...

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**1**answer

168 views

### Naive question on the Jacobian of a curve

Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the ...

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**1**answer

256 views

### Degeneration of smooth curves and Picard-Lefschetz formula

Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...

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**3**answers

290 views

### Smoothen a nodal curve

Let $k$ be a field, let $X/k$ be a nodal curve (which means $X_{\overline{k}}$ is a connected reduced proper curve with at worst nodal singularities.)
Does there always exists a proper flat family $\...

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**1**answer

148 views

### Why are modular curves non-trivial covers of the $j$-line

This is a very soft question.
Let $n\geq 1$ and let $Y(n)$ be the (open) modular curve associated to $\Gamma(n)\subset SL_2(\mathbb{Z})$. Interpreted correctly, $Y(n)\to Y(d)$ is finite etale, ...

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181 views

### Is a torsion free sheave of rank one on a reducible curve the pushforward of a line bundle on a normalization?

Let $X$ be a nodal curve, possibly reducible. Then can any torsion free sheaf of rank one on $X$ be expressed as $\pi_*(L)$, where $L$ is a line bundle on a partial normalization of $X$? This looks ...

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**1**answer

179 views

### An explicit correspondence for reductions of modular curves $Y(N)$

Let $Y(N)$ be the modular curve associated with the principal congruence subgroup $\Gamma(N) \subset \mathrm{SL}(2, \mathbb{Z})$ of level $N \in \mathbb{N}$. It is well known that this curve has a ...

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**1**answer

126 views

### Do line bundles with enough sections on surfaces have generic divisors which are irreducible?

Let $L$ be a line bundle on a smooth connected complete complex algebraic surface $X$. Assume that $L$ has enough sections i.e. that $H^0(L,X)$ has dimension $> 1$. A nonzero section $s$ of $L$ ...

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**1**answer

462 views

### Faltings theorem and number of singularities

The Faltings theorem states that the number of rationals over an algebraic curve is finite if the genus is greater than 1. The genus decreases by increasing the number of singularities. My question is ...

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96 views

### Pascal theorem for three dimensions

A year ago I found the Pascal theorem for three dimentions as follows:
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...

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**1**answer

213 views

### Some curves on the Jacobian of a genus $2$ curve and their image under certain maps (char $p$)

I hope this question belongs here. The situation in this question is quite particular and specific.
I am trying to weak some of theory to measure the degree of some function on the Jacobian of a ...

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245 views

### An example of curves with the same Jacobian, but different Jacobian automorphism groups (wrt their respective canonical principal polarizations)?

I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian.
Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...

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425 views

### Priority for lemniscate of Gerono?

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). There seems to have been ...

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85 views

### Tangent Bundle of reducible genus one curves

I need to know what can be said in general about the tangent bundle of reducible curves over complex numbers with arithmetic genus one, say $I_N$.
As far as I know for any Simpson semistable torsion ...

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86 views

### Intersection number of two projective curves using the resultant and tangent lines

For my thesis, I'm working on the intersection of projective plane curves over $\mathbb{C}$. We define the intersection number of projective plane curves (see for example Gibson - Elementary Geometry ...

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112 views

### Embedding of curves in $\mathbb P^2$

There is a mistake in the following argument but I cannot see where. Can someone help me, please?
Let $C$ be any smooth curve of genus $g\geq 1$ and $D$ a general effective divisor of degree $g+2$. ...

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210 views

### Endomorphism of involution invariant vector bundles

Let $C$ be an hyperelliptic curve with an involution $\sigma$, and let $E$ be a rank $2$ stable involution invariant vector bundle on $C$, that is $\sigma^{*}E \cong E$. Let $(End(E) \otimes K_C)_{+}$ ...

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**1**answer

117 views

### Variation of global sections of line bundles

The underlying field is $\mathbb{C}$.
Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\...

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**1**answer

164 views

### Moduli problem of stable nodal curves over the integers

Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}$ represents the functor $$\overline{\mathcal{M}}_{g,n}(...

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**1**answer

159 views

### An explicit computation of the blow-up of curve over $\mathbb{F}_3$ at two points

I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are ordinary double points.
Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 ...

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72 views

### On a presentation of $\bar{M}_{2,1}$

In this answer it is said that $\bar{M}_{2,1}\cong \bar{M}_{0,7}/S_6$. However, I cannot see this. Given a curve of genus $2$ and a marked point, quotienting by the involution surely gives a rational ...

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102 views

### Uniqueness of Mukai presentation of canonical model in genus 6

In his 92 paper, Mukai showed that a general genus $6$ curve may be represented in $\mathbb{P}^9$ as the intersection of the Grassmannian $G(2,5)$ (under the Plucker embedding), a plane $H\cong \...

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**1**answer

291 views

### surface with rational curve in the double locus

I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist):
$X$ is slc (and not-normal)
There is rational curve $C \...

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**2**answers

245 views

### A log structure on the moduli space of curves

Let $M_{g, n}$ be the moduli space of curves of genus $g$ with $n$ marked points. Let $M_{g, \vec{n}}$ be the moduli space of marked curves with a choice of a (possibly zero) tangent vector at each ...

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100 views

### Torsion line bundle on hyperelliptic curves and Weierstrass points

Let $C$ be an hyperelliptic curve of genus $g$ and let $f:C\rightarrow
\mathbb{P}^1$ be the corresponding 2 to 1 covering
ramified in $2g+2$ points.
Let $L$ be a line bundle on $C$ such that either $...

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93 views

### Structure of Derived Category of Sheaves on a curve

Let $C$ be a smooth projective curve over $\mathbb{C}$. We will be working in the topological category. Let $A \in D^b(C):=$ bounded derived category of sheaves on $C$. Is it true that $$A \cong \...

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203 views

### Natural model for genus $6$ curves

By Brill-Noether theory, the generic genus $6$ curve is birational to a sextic plane curve in $\mathbb{P}^2$. I was wondering if there is a direct/natural construction of this birational map. In other ...

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40 views

### How can I describe in explicit geometric terms the (in general non-complete) linear system?

Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\...

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120 views

### On a smooth curve $C$, when is $K_C \sim_\mathbb{Q} (2g-2)P$?

Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:
There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)...

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85 views

### Twisting a line bundle with the zero section

Let $X$ be a smooth projective curve and $L$ be an invertible sheaf on $X$. Denote by $\mathbb{L}$ the line bundle associated to $L$, $\pi:\mathbb{L} \to X$ the natural morphism and $0_\pi$ the zero ...

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**2**answers

183 views

### Image of curve along a finite etale Galois map

Let $f:X\to Y$ be a finite etale Galois morphism of varieties over $\mathbb{C}$. Let $C$ be a smooth quasi-projective connected curve in $X$.
Is $f(C)$ a smooth curve?

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52 views

### Is the isogeny class 1109.a of abelian surfaces in the LMFDB complete?

The LMFDB lists the Jacobian of the genus-2 curve 1109.a.1109.1 (http://www.lmfdb.org/Genus2Curve/Q/1109/a/1109/1) as being isolated in its rational-isogeny class. However, the LMFDB does not purport ...

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248 views

### Deformation of “Hecke modification”

Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy:
...

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108 views

### Descent via an explicit isogeny (genus 2)

This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians.
Here I ask some technicalities of a ...

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223 views

### Rational normal curves and tangent lines

Let $C,\Gamma\subset\mathbb{P}^n$ be degree $n$ rational normal curves in $\mathbb{P}^n$, such that for any $p\in C$ the tangent line $T_pC$ of $C$ at $p$ is tangent to $\Gamma$ as well. This means ...

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**1**answer

215 views

### A map from a symmetric product of a curve to its Jacobian

Let $C$ be a smooth projective curve over an algebraically closed field $k$, of genus $g$.
It is well known that, after fixing a point $p_0$, the map $C^{(n)}\to J$ sending $\{a_1,\dots,a_n\}$ to $[...

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**1**answer

245 views

### Descent of coherent sheaves on finite coverings

Let $X$ be a non-singular hyperelliptic curve (over $\mathbb{C}$) and $\pi:X \to \mathbb{P}^1$ be a $2:1$ covering. Let $\sigma:X \to X$ be the hyperelliptic involution and $E$ be a locally free sheaf ...

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167 views

### Non-reduced curve of arithmetic genus $0$

Let $C$ be a $1$-dimensional proper finite type scheme over field $k$, such that $p_a(C)=0$ (or equivalently $h^0(\mathcal{O}_C)-h^1(\mathcal{O}_C)=1$), then is it true that the reduced curve $C_{red}$...

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213 views

### What is the kernel of $i^*:H^*(\overline{\mathcal{M}}_{g,n},\mathbb{Q}) \to H^*(\overline{\mathcal{M}}_{g-1,n+2},\mathbb{Q})$?

Let $\overline{\mathcal{M}}_{g,n}$ be the Deligne-Mumford moduli space of stable algebraic curves. I would like to know which rational cohomology classes or at least which tautological classes on $\...

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**1**answer

244 views

### Rational curves in ${\mathbb P}^n$ and immersion

In the paper of Herbert Clemens
Curves on generic hypersurfaces
the author shows that for a generic hypersurface $V$ of ${\mathbb P}^n$ of sufficiently high degree there is no rational curve on $V$...

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**1**answer

196 views

### The degree of the Gauss map of Theta divisor

Let $R$ be compact a Riemann surface of genus $g$ and $ J (R) $ be its Jacobian.
For a subvariety $X$ of $J(R)$ of dimension $d$, denote the set of non-singular points of $X$ by $X_{reg}$. Then the ...

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**1**answer

538 views

### Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

We work over the field of complex numbers. (But remarks in characteristic $p$ are very welcome.)
Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains ...

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**3**answers

319 views

### Meaning of general hyperplane $H$ in $\mathbb{P}^n$

Maybe this question is not suitable for this platform, I already put that same question in math.stackexchange and I find only vague answers.
I'm studying the book of Rick Miranda; Algebraic Curves ...

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**1**answer

274 views

### The set of points of the fiber product of two algebraic stacks and the fiber product of the sets of points of two algebraic stacks

Let $\overline {\mathcal{M}_{g_{i}, n_{i}}}, \ i \in \{1, 2\},$ be a moduli stack of pointed stable curves of type $(g_{i}, n_{i})$ over a finite field $\mathbb{F}_{p}$. For any algebraic stack $\...