# Questions tagged [algebraic-curves]

An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

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### Doubt in a proposition from Fulton's Algebraic Curves, section 5.5 (some criteria for Noether's condition)

I was reading section on Max Noether’s Fundamental Theorem in Fulton's Algebraic Curves and came across the following proposition which gives some sufficient criteria for Noether's condition to hold (...
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### Does maximal contact hypersurface (m.c.h.) exist for any curve over a field of arbitrary characteristic, and if so, for what definition of m.c.h.?

Does a maximal contact hypersurface always exist for any curve over a field of arbitrary characteristic, and if so, for what definition of maximal contact hypersurface?
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### Order of Picard groups of non-hyperelliptic algebraic curves

Let $q$ be prime. When $E/\overline{\mathbb{F}_q}$ is an elliptic curve, it is well-known that the group of $\mathbb{F}_q$-points of $E$ is isomorphic to the Picard group of degree $0$ divisors on $E$ ...
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### Smoothing projective nodal curve, is the general fiber smooth?

Proposition 29.9 of Hartshorne's Deformation theory states the following: A reduced curve Y in $\mathbb{P}^n$ with locally smoothable singularities and $H^1(Y,O_Y(1)) = 0$ is smoothable. In particular,...
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### What is a curve at $y=\infty$ mean?

On the wikipedia for the trident curve, $xy+ax^3+bx^2+cx=d$, two graphs are shown: Both are for the case where $a=b=c=d=1$, with the first matching what I find in desmos, but the latter being the '...
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### Understanding Gauss-Manin connection on Arbarello's book

I am trying to understand the Gauss-Manin conecction in order to understand the definition of the Period Mapping on the Moduli space of algebraic curves of genus $g$ and its extension to the ...
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