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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.

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.

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How does the failure of integral closure of a coordinate ring relate to the cusp/singularity we see in the corresponding variety?

Consider the affine variety with coordinate ring $R=\mathbb{C}[x,y]/(y^2-x^3)$. It is clear that the coordinate ring $R$ is not integrally closed since $t=y/x$, an element of the corresponding quotient field, is a root of an integral equation.…
user7090
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Proof that a Zariski closed subset of a projective space is the common zeros of homogeneous polynomials

Let $K$ be an algebraically closed field. Let $n \ge 0$ be an integer. We consider $K^{n+1}$ as a topological space with Zariski topology. Let $G = K^*$ be the multiplicative group of $K$. Let $X = K^{n+1} - {0}$. Then $G$ acts on $X$. Then the…
Makoto Kato
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Intersection multiplicity of two curves.

I want to compute the intersection multiplicity of $YZ=X^2$ and $YZ=(X+Z)^2$ at $P=(0:1:0)$ In an affine nbd of $P$, let $(X:Y:Z)=(x:1:z)$ $$I_P=\dim \mathcal{O}_{\mathbb{A}_k^2,(0,0)}/(x^2-z,x^2+2xz+z^2-z)$$ In my previous question, I learned that…
Gobi
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Tangent Spaces and Morphisms of Affine Varieties

In page 205 of "Algebraic Curves, Algebraic Manifolds and Schemes" by Shokurov and Danilov, the tangent space $T_x X$ of an affine variety $X$ at a point $x \in X$ is defined as the subspace of $K^n$, where $K$ is the underlying field, such that…
Manos
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The shape of elements of schemes

In the book Geometry of Schemes, written by David Eisenbud & Joe Harris, at the start of chapter one in the section Schemes as Sets, the authors introduce elements of $R$ as functions. I can't understand their meanings there! If anyone knows how…
someone
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What is an Affine Cone?

When looking at a projective variety, we can intersect the variety with the standard affine patches and the union of these intersections give the projective variety. So the variety can be viewed as a projective variety over $\Bbb P^n$ or as an…
Nivedita
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$G$- invariant part of push forward

Let $C$ be a smooth projective curve. Let $G$ be a finite group which acts on $C$. Let $C'=C/G$ the quotient of the action, which is a smooth curve. Then $f:C\rightarrow C'$ is a finite, possibly ramified morphism. 1) Let $L$ be a line bundle on $C$…
user52991
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How to Compute Projective Closure in General?

There have been a few questions about this on this site, but I think my question is different because a) my question isn't about Hartshorne 2.9, it's just inspired by that question, and b) the other questions don't ever seem to actually describe how…
A. Thomas Yerger
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Pole set of rational function defined on a variety

The problem: Let $V = V(y^2-x^2(x+1))$, and let $\overline{x}, \overline{y}$ denote the $I(V)$-residues of $x$ and $y$ in the coordinate ring $\Gamma(V)$. Set $z=\overline{y}/\overline{x}$. Find the pole sets for $z$ and $z^2$. My progress: Since…
Ekie
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Compute a canonical divisor

Consider $C,C'$ two cubics in the plane with $C$ smooth. There are $9$ basepoint on the linear system generated by $C$ and $C'$ so if we blow them we get a map $X \to\mathbb P^1$, where $X$ is $\mathbb P^2$ blown-up at 9 points. Now is my question :…
user378546
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Example of the fixed point of a linear system?

Let $X$ be a normal projective variety and $D$ a Cartier divisor on it. A point $p\in X$ is called a fixed point of $|D|$ if $p \in \operatorname{supp}(D')$ for any $D'\in |D|$. Here $|D|$ is the linear system of $D$ defined by…
M. K.
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Canonical class of a singular variety

On a normal variety (possibly singular) one can make sense of the canonical class $K_X$ by computing it on the smooth locus and then extending over the singular part. I'm wondering how to actually do this in practice: for example, on the singular…
user6626
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Surjective regular morphism from affine space to punctured plane

Does there exist $d$ and a regular (=polynomial) map from the affine space $\mathbb{A}^d$ to $\mathbb{A}^2$ whose image is exactly the punctured plane $\mathbb{A}^ 2\smallsetminus\{0\}$? Here the base field is algebraically closed, and of…
YCor
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Relationship Between the Albanese Variety and $\rm{Pic}^{0}(X)$

I've been learning about the Albanese variety $\rm{Alb}(X)$ of a projective variety $X$. As discussed very well in John Baez' blog (https://golem.ph.utexas.edu/category/2016/08/the_magic_of_algebraic_geometr.html) there is apparently an analogy…
Benighted
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Image of the Veronese Embedding

Is the image of the general Veronese embedding ever contained in a hyperplane of $P^{n}$? I'm guessing no, but I can't prove it.
Abelsh
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