# Tagged Questions

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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### Flat families of semistable sheaves parametrized by $\mathbb{A}^1$.

Suppose we have a non trivial short exact sequence, $$0\longrightarrow F'\longrightarrow F\longrightarrow F''\longrightarrow0,$$ where $F$, $F'$ and $F''$ are semistable sheaves with the same reduced ...
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### Is there an algebraic-geometric solution to the problem of the Leibnizian formalism?

The precise question appears at the end of this entry. With all the recent advances in understanding infinitesimals, we still don't fully understand why Leibniz's definition of $\frac{dy}{dx}$ as ...
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### G-equivariant invertible sheaves on affine curves

Let $A$ be a Noetherian integral domain, and $G$ a finite group of automorphisms acting on $A$. Let $B = A^G$, the ring of invariants. The inclusion $B \hookrightarrow A$ induces a surjective morphism ...
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### Why is this theorem important in fulton's book?

In Fulton algebraic curves book, we have the following proprieties which help us to find the intersection number of a pair of function at a given point. afterwards he states this theorem: What'...
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### parametrization of a specific algebraic surface

I consider the surface of degree $5$ of equation: $$4 x y - 4 x^3 y - 4 x y^3 + z + 2 x^2 z + x^4 z + 2 y^2 z + 2 x^2 y^2 z + y^4 z - 4 x y z^2 - 6 z^3 + 2 x^2 z^3 + 2 y^2 z^3 + z^5=0$$ Question: ...
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### What are the prerequisites for Fulton's “Intersection Theory”?

Is it necessary to read SGA VI to understand "Intersection Theory" by William Fulton?
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### $x^2+y^2=z^2$ in complex numbers

as a prelude to inquiring about solutions of Pythagoras' equation in Gaussian integers, it seemed sensible first to write out this equation for the complex case! i use the notation $z_i=x_i+iy_i$ and ...
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### How to prove this comment of Fulton

I'm trying to understand why this is true in Fulton's Algebraic Curves: Why we add this point $(0,\ldots, 0)$? Why this equality is true? I really need help. Thanks in advance.
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### Thomason resolution of sheaves

Let $X$ be a smooth quasi-projective scheme over a field $k$ and $G$ an algebraic group (also over $k$ not necessarily reductive) acting on $X$. I the work of Thomason "Equivariant Resolution, ...
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### Finite surjective morphism of smooth varieties is flat

Let $f: X \to Y$ be a finite surjective morphism of nonsingular varieties over a field $k$. Exercise III 9.3. in Hartshorne's Algebraic Geometry sais that if $k$ is algebraically closed, then $f$ is ...
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### Finding an algebraic equation given divisors

I'm trying to find an algebraic curve that represents a specific Riemann surface and my question goes like this: Given divisors $(\omega_1) = P_1 + 5 P_2 + 2 P_3,$ $(\omega_2) = 5 P_1 + P_2 + 2 P_3,$ ...
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### Homology of non-singular projective algebraic variety

I am unsure whether or not the following claim is true or false and whether or not my proof works or not: Claim: Let $V \subset \mathbb{C}P^n$ be a complex $k$-dimensional, non-singular, projective ...