Mathematics Stack Exchange
Mathematics Stack Exchange

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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Finding the Transformation to a Canonical form for a Quadric Surface

I am attempting to calculate the intersection of quadric surfaces defined by $0 = X^T A X$, $0 = X^T B X$, $0 = X^T C X$ with X = [x, y, z, 1]. Matrices A, B and C are real and symmetric. There are several methods for solving, most of which trace…
Damien
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Finite Fibers over closed points

Let $f:X \rightarrow Y$ be a morphism of algebraic varieties over an algebraically closed field. If all fibers $f^{-1}(y)$ with $y$ closed point in $Y$ are finite, can one conclude that an arbitrary fiber (i.e. with $y$ not necessarily closed point)…
Cyril
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Scheme flat of finite type over $\mathbb{Z}$

Let $X$ be a scheme which is integral, of finite type, flat and separated over $\mathbb{Z}$. Let $D \subseteq X$ be a prime divisor on $X$ which is not flat over $\mathbb{Z}$. Is it true that $D(\mathbb{F}_p) = \emptyset$ for all primes $p$, with…
Evariste
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Is a bijective morphism of quasi-affine smooth varieties an isomorphism?

We have a bijective morphism $f:X\to Y$ of quasi-affine varieties (say over $\Bbb{C}$). Can $f$ fail to be an isomorphism if $X$ and $Y$ are smooth?
Heitor Fontana
  • 3,676
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Line Bundle of deg $2g-1$ and generated by global sections

Let $X$ be a smooth projective curve of genus $g \geq 2$ over $\mathbb{C}.$ Does there exist a line bundle $L$ on $X$ of degree $\deg L= 2g-1$ such that it is generated by global sections? (One can show that if $L$ is a line bundle of degree $\deg…
Suhas
  • 497
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Pullback divisor as a preimage scheme?

If $D$ is an effective divisor on a curve $Y$, then I can represented $D$ as a zero dimensional quasi-compact subscheme: for a closed point of multiplicity $n \geq 0$, we have a closed immersion from $Spec k[x] / (x^n)$ onto $Y$. Suppose that $f :…
Elle Najt
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rationally connected by a smooth curve

Let $X$ be a smooth, rationally connected variety over $\mathbb{C}$. A priori, this means that for any two points $p,q\in X$, there is a rational curve $C$ containing $p$ and $q$. Can $C$ be taken to be smooth?
adrido
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Computing degree and ramification indices for given morphism of irreducible smooth curves in $\mathbb P^2$

Let $V = Z(X_0^8 + X_1^8 + X_2^8) $ and $W = Z(X_0^4 + X_1^4 + X_2^4)$ in $\mathbb P^2$. It can be shown that $V$ and $W$ are irreducible curves. Let $\phi : V \to W, (X_i) \mapsto (X_i^2)$ be a morphism. I want to show that $V$ and $W$ are smooth,…
Jonathan
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Generic fiber of a scheme over a DVR

What is (usually?) meant by the generic fiber of a scheme over a discrete valuation ring? I've seen in some talks now, could somebody give a precise definition? Thank you very much in advance!
asdfasgadsf
  • 53
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Function fields of irreducible varieties

The following might seem long, rambling and to contain more information than necessary. The problem is, I'm having a macro-understanding issue and feel like I need to tell you everything I think so that you can tell me why I'm wrong. I'm learning…
Jonathan
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Product of affine schemes

For any ring $A$, define a functor $\text{Spec}(A)$ from rings to sets by $$\text{Spec}(A)(R) = Hom_{\text{Rings}}(A,R)$$ Call a functor $X$ an affine scheme if it is isomorphic to a functor of the form $\text{Spec}(A)$. The Yoneda lemma says that…
Juan S
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references for singularities: does quotient singularities imply gorenstein?

Is there a good place where to learn about singularities of algebraic varieties? OK, this question is terribly vague, so I'll ask what I'm currently interested in: if X is a smooth variety and G is a finite group, then is $X/G$ Gorenstein? If true,…
Jacob Bell
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Construction of Projective Varieties Question

I am struggling in understanding Mumford's construction of Projective Varieties. In the image I uploaded here, Are we to understand each $R_n$ as $M_n/P_n$, where $M_n:=${homogeneous polynomials in $k[x_1,...,x_n]$ of degree n} and…
kfriend
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Sections of ruled surfaces

The following questions maybe elementary, but I can't find them in the literature. Assume now everything I will write is defined over some algebraically closed field. Let $S$ be a (geometrically) ruled surface over a nonsingular projective curve $C$…
user41541
  • 323
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On the definition of degree of a hypersurface

Let $f \in \mathbb{C}[x_1, ..., x_n]$ be a homogeneous polynomial of degree $d$. I was trying to understand the definition of degree of hypersurfaces. It says on Wikipedia https://en.wikipedia.org/wiki/Degree_of_an_algebraic_variety that "The…
Johnny T.
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