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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.

29074 questions
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Why to use such a complex definition of intersection multiplicity?

Let $X$ be a smooth variety and $V, W$ two closed irreducible and reduced subvarieties represented by ideal sheaves $I$ and $J$. Serre defines an intersection multiplicity for an irreducible component $Z$ of $V\cap W$…
Alex
  • 221
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Elements of Spec$(\mathbb{C}[x_1,\dotsc,x_n]/(f_1,\dotsc,f_r))$.

I was reading in Vakil's Foundations of Algebraic Geometry that one can picture the "traditional points" of Spec($\mathbb{C}[x_1,\dotsc,x_n]/(f_1,\dotsc,f_r))$ as the zero locus of the polynomials $f_1,\dotsc,f_r$. (This is early in the book, and I…
Erik
  • 135
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Question about twisting sheaves: Twisting by two points on $\mathbb{P}^1$ and basepoints of $\mathscr{O}(-1)$

I was trying to do some computations to get familiar with twisting sheaves and I ran into some questions. I would greatly appreciate any help! Thanks in advance! For line bundles, being globally generated is the same as being basepoint free. The…
user153327
  • 53
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Normalisation of curve

I am probably having a lot of confusion with the terminologies in shafarevich. In page 131, Normal varieties, it states a corollary. An irreducible algebraic curve is birational to a nonsingular projective curve. Now I can't find "algebraic curve" …
Daniel
  • 53
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Hartshorne Propositon I.3.3

In Hartshorne book Proposition (I.3.3) is that Proposition : Let $U_i\subset \mathbb{P}^n$ be the open set defined by the equation $x_i\neq 0.$ Then the mapping $\varphi_i : U_i\longrightarrow \mathbb A^n$ is an isomorphism of varieties. Let…
Med
  • 1,598
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Hyperplane not containing a given set of points over a Noetherian scheme

This is related to the answer in this question: Showing that a power of an ample sheaf is equivalent to an effective Cartier divisor Let $X$ be a quasiprojective scheme over a Noetherian ring A and suppose we have a very ample sheaf $\mathcal{L}…
user101036
  • 818
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Cartier divisors and global sections

I have a brief question - I seem to have a vague recollection that if we have a Cartier divisor $D$ on a scheme $X$ , then we can determine whether $D$ is effective by saying whether $\mathcal{O}_X(D)$ has a global section or not. I have tried to…
user101036
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Every open subset of an affine variety is itself an affine variety

It seems that I have proved the title claim, but I'm not convinced by my own argument, so can you check where is the error, if one? Let $X\subset\mathbb{A}^n$ be an affine variety and let $Y$ be an open subset of $X$. I want to prove that $Y$ is…
Danae Kissinger
  • 2,143
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d-uple embedding

When one restricts the $d$-uple embedding $\mathbb{P}^n \hookrightarrow \mathbb{P}^N$ to $\mathbb{P}^{n-1} \hookrightarrow \mathbb{P}^n$, does this yield the $d$-uple embedding $\mathbb{P}^{n-1} \hookrightarrow \mathbb{P}^N$? The degree of…
user5262
  • 1,863
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Normalization bijective on smooth points?

Suppose we take an algebraic variety $X$ over $\mathbb{C}$ (I assume reduced). Is the normalization $$\pi:\tilde{X} \to X$$ always bijective on the smooth points of $X$?
Karsten
  • 233
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Fixed points of the torus action on $\textrm{Hilb}_n(\mathbb C^2;d)$

On the affine plane $\mathbb C^2$ we have the action of the torus $T=(\mathbb C^\times)^2$ given by rescaling: $$(t_1,t_2)\cdot (a,b)=(t_1a,t_2b)\in\mathbb C^2.$$ This action extends to the Hilbert scheme $\textrm{Hilb}_n(\mathbb C^2;d)$ consisting…
Brenin
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Semistable vector bundles elliptic curve

Let $n=(r,d)$, r=r'n, d=d'n and $M(r,d)$ the moduli space of $S-$equivalence classes of semistable bundles of rank $r$ and degree $d$. How can I construct a finite morphism $M(r',d')^n\to M(r,d)$ equivariant for the simmetric group $S(n)$ and such…
dolce
  • 301
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How to determine a set of polynomials is algebraically indepedent or not?

I have a set of polynomials $$ a a_1, bb_1, c, c_1, ab, da_1 + bd_1, fa_1+cf_1+d_1e, eb_1+ce_1, de-bf, f_1, e_1, d_1. $$ Is there some software which can determine that they are algebraically independent or not. If dependent, how to find the…
LJR
  • 14,520
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If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$?

The following is the Proposition 1.10 of Hartshorne's Algebraic Geometry: If $Y$ is a quasi-affine variety, then $\dim Y =\dim \overline{Y}$. In the proof, there is a statement saying that if $$ Z_0 \subset \cdots \subset Z_n \;\;\; (1) $$ is a…
Aki
  • 1,299
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The Affine Tangent Cone

I'm failing to see how exactly is the tangent cone at a singular point on a curve picking out all the different tangent lines through this singular point (say the origin in $\mathbb{A}^2$)? Could someone explain this, or at least redirect me to a…
V-B
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