Questions tagged [algebraic-geometry]
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. This tag should not be used for elementary problems which involve both algebra and geometry.
31,096 questions
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On the dimension of a linear system to give an embedding in $\mathbb{P}^3$
On p. 353 in section IV.6 of Hartshorne's Algebraic Geometry, Hartshorne poses the question of whether a curve of degree $7$ with $g=5$ exists in $\mathbb{P}^3$. He then says
We need a very ample ...
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Rank $2$ bundle on a smooth projective variety
Let $E$ be a rank $2$ bundle on a smooth projective surface. Then we know that $E^{*} \otimes \text{det}(E) \cong E$. Does this still hold if the dimension of the variety is strictly bigger than $2$ (...
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Using geometric constructions to solve algebraic problems (in Euclid and Descartes)
Long-time listener, first-time caller here. I've got what I guess is a history of math question. I've been teaching a college course on Euclid's Elements (surprisingly fun, btw). I've been using David ...
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Meaning of "nilpotents on the a curve" in Hartshorne.
I just wanted to check my understanding of a phrase in Hartshorne, and the following calculations. In example II.3.2.4, we consider $\mathrm{Spec}(k[x, y])$ and consider the ideal $(x)^2$ with the ...
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Exercise I.3.19, Hartshorne - still an open problem?
For completion, I here state what Exercise I.3.19 in Hartshorne "Algebraic Geometry" says
Let $\varphi:\mathbb{A}^n \to \mathbb{A}^n$ be a morphism of $\mathbb{A}^n$ to $\mathbb{A}^n$ given ...
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Canonical divisor of a singular surface.
Let $S$ be a normal projective algebraic surface over a complex numbr field ${\Bbb C}$. Suppose that $S$ has a single isolated singular point $p \in S$. Let $U \colon= S\setminus p$ be a smooth open ...
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Understanding a proof in Milne's notes on algebraic geometry
In his notes on algebraic geometry,Milne defines an algebraic variety as a prevariety with a separated condition:
A prevariety $V$ is separated if for all regular maps
$\varphi_1,\varphi_2:Z\...
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When does a non-hyperelliptic curve allow a hyperelliptic cover?
Let $C$ be a nice (i.e. smooth, projective, and geometrically integral) curve of genus $g \geq 3$, defined over some field $k$. I am mainly interested in the case that $k$ is a number field, but feel ...
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How do I know if this polynomial in several variables is irreducible? [closed]
How do I know if this polynomial is irreducible?
$x^2 − y^2 z^2 - z^3 $
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Riemann Surfaces to Graphs
In the paper Riemann-Roch and Abel-Jacobi Theory on a Finite Graph by Matthew Baker and Serguei Norine, it is mentioned that 'It is well-known that a finite graph can be viewed, in
many respects, as a ...
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Inflection point on elliptic curves
I've been having trouble coming up with a full-fleged answer for a question in a homework assignment for my algebraic geometry class. The question says:
Let $F$ be an elliptic curve (projective cubic ...
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Boundary map for Schubert cells
I am learning about real Grassmannian.
We define the Schubert cell $e(m_1,...,m_s)$, so $m_j=m_j(\pi)=\inf\{m\mid \dim(\pi\cap \mathbb{R}^m)\geq j\}$ for $1\leq j\leq k$, where $\pi$ is a $k$-plane in ...
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Localization at a prime of a subring preserves Noetherianity
Let $B \subseteq A$ be an extension of Noetherian rings, and let $\mathfrak{p} \trianglelefteq B$ be a prime ideal. The localization of $A$ (viewed as a $B$-module) at $\mathfrak{p}$ is $A \otimes_B ...
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Down-to-earth definition of "genus".
Cubic Curve to Weierstrass Form
For the cubic curve $C$ in general form with rational coefficients:$$ax^3+bx^2y+cxy^2+dy^3+ex^2+fxy+gy^2+hx+ky+l=0,$$we are interested in finding rational points on it. ...
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Vakil 6.5.J on decomposition of finite length modules
Let $M$ be a finite length $A$-module, with composition series $0 = M_0 \subset M_1 \subset \cdots \subset M_n = M$. We can write $M_i/M_{i-1} = A/\mathfrak{m}_i$ for some max ideal $\mathfrak{m}_i \...
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