# Questions tagged [algebraic-curves]

for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.

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### Smooth proper local model of a smooth projective curve

Say I have a curve $C/K$, where $K$ is a number field. Let $v$ be a place of $K$, and denote by $K_v$ the $v$-adic completion of $K$. Further assume $C$ is smooth and proper over $K$. Denote by $C_v$ ...
64 views

### Smooth proper models of hyperbolic curves

Let $C$ be a curve defined over a number field $K$. I am interested in knowing under which conditions on $C$, does it have a smooth proper model? I understand that when $C$ is a smooth, projective, ...
75 views

### Properties of the symmetric square of a curve [migrated]

Under what conditions on a curve $C$, defined over a ring $R$, is its symmetric square, $C^{(2)}$ smooth/proper? Is it enough for $C$ itself to be smooth/proper over $\text{Spec}R$? How does one see ...
115 views

### Embedding of symmetric square in Jacobian

Let $C$ be a projective curve defined over a field $K$, and let $C^{(2)}$ and $J$ be its symmetric square and Jacobian, respectively. There is a natural map $C^{(2)}\hookrightarrow J$, defined as ...
1 vote
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### Canonical basis of cycles of Riemann surfaces

Let $\Gamma$ be the compact Riemann surface defiend by the algebraic curve $$f(x, y) = y^n + a_1(x)y^{n-1} + a_2(x) y^{n-2} + \dots + a_{n-1}(x)y + a_n(x) = 0,$$ where $a_1(x), \dots, a_n(x)$ are ...
111 views

### Integration on algebraic curves

Consider the plane algebraic curve $$f(x, y) = y^4 - (2x - 1)y^2 - (4x - 1) y + x^2 + x + 1 = 0.\tag{1}$$ Its compactification results in a Riemann surface $C_1$ of genus $1$. Hence, it can be ...
292 views

### Existence of space curves of given genus and degree

In Hartshorne's Algebraic Geometry Chapter IV, Section 6, he summarizes known results on the existence of smooth space curves of degree $d$ and genus $g$ for $g\le 12$ and $d \le 10$. He shows the ...
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94 views

### Degeneration differential form nodal curve

I have a (possibly very basic) question about differential forms on nodal curves. After reading Witten's survey "Two-dimensional gravity and intersection theory on moduli space", I am ...
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Let $C$ be a curve of large genus $g > 1$ over an algebraically closed field of characteristic $0$, and let $G = \textrm{Aut}(C)$ be its automorphism group. Is there a general way to compute the ...