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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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Existence of a Cech cover for computing Picard group

Let $X$ be a variety -- one can compute $\text{Pic}(X) = H^1(X, \mathcal{O}^*_X)$ by choosing a Cech cover which is acyclic with respect to $H^\bullet(-, \mathcal{O}^*)$. Can one always do this? It seems to me that the answer is no. For example,…
user148177
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Basic question regarding degrees of algebraic sets

I am learning about degrees of algebraic sets at the moment, and in an article I am reading I came across the following: Let $V_i \subseteq \mathbb{C}^n$ be a hypersurface of degree at most $D$ for each $1 \leq i \leq l$. Let $V = \cap_{1 \leq i…
Johnny T.
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Isomorphism of Varieties

Let $V=V(x^2+y^2-1) \subset \mathbb{R}^2$ be an affine variety. Show that $V$ is rational, but isn't isomorphic to $\mathbb{R}^1$. I could show that $V$ is rational, by parametrization $$x=\frac{1-t^2}{1+t^2} , y=\frac{2t}{1+t^2}, t \in…
Cgomes
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Formula for top self intersection of exceptional divisor

Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Let $\pi: Y \to X$ be the blow-up of a smooth point $x \in X$. Is there a nice formula for the intersection number $E^n$?
Evariste
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Correspondence between infinitessimal extensions and cohomology

I am trying to understand the correspondence between isomorphism classes of infinitessimal extensions of a k-scheme $X$ by a coherent sheaf $\mathcal{F}$ and the cohomology group $H^1(X,\mathcal{F}\otimes \mathcal{T}_X)$ where $\mathcal{F}_X$ is teh…
Luc
  • 751
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Deriving the Quadratic Polynomials Defining the Twisted Cubic

I've recently been reading about rational normal curves and how they may be represented and have come to the following question: (for simplicity's sake, the problem is stated in terms of the RNC in $\mathbb{P}^3$, i.e., the twisted cubic) The…
aherring
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Irreducible curve contained in linear subspace

Can someone give me a starting point for the following question? I don't know where to begin! Let $C \subset \mathbb{P}^n$ be an irreducible curve of degree $d$. Show that $C$ is contained in a linear subspace of $\mathbb{P}^n$ of dimension $d$.
TheBeiram
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Intersection of ample and effective divisors

I believe it is something silly, but I'm a newbie, so why on a surface the intersection of an effective divisor and a divisor from ample bundle is non-negative? In fact, I need that an intersection of an effective divisor with a hyperplane section…
evgeny
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27 lines on Fermat surface

I want to describe the $27$ lines on the Fermat surface and have found the information below. I don't understand the last part. How is it possible to go from $9$ different lines to $27$ different lines by changing the projective coordinates. Can…
Karen
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Two questions about the proof of Proposition V.2.2 in Hartshorne's Algebraic Geometry

This proposition on page 370 is to prove any ruled surface over a nonsingular curve $C$ is $\bf{P}(\mathscr E)$, the projective space bundle of a locally free sheaf $\mathscr E$ of rank $2$ on $C$ and vice versa. In the direction that for a given…
user41541
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Points of scheme with residue field $k$ vs $k$-point

Let $X$ be a scheme over a field $k$. Consider the following definitions. The residue field of a point $x\in X$ is $k(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x$. The $k$-point of $X$ is the morphism of schemes $\text{Spec}\,k\to X$ such that the…
vitaliy
  • 577
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Finding irreducible components of a closed set

Let $k$ be a field with char(k) $\neq 2$. How can we decompose into irreducible components the following set: $Z(x^{2}+y^{2}+z^{2},x^{2}-y^{2}-z^{2}+1) \subseteq \mathbb{A}^{3}$? Doing the algebra leads to $x^{2}+\frac{1}{2}=0$ and…
user6495
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Exercise 2.12 in Harris - Algebraic Geoemetry: a first course

Consider the three lines of $\mathbb{P}^3$ given by $L: \, z_0 = z_1 = 0 \\ M: \, z_2=z_3 = 0 \\ N: \, z_0 = z_2, \, z_1 = z_3.$ It is claimed in Exercise 2.12 of Harris (a first course) that the union of all lines that meet all $L,M,N$ is…
Manos
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Nonsigular curve of degree $3$ in $\mathbb P^2$ over a field of characteristic $3$

I am trying to do problem $1.5.5$ from Algebraic Geometry by Robin Hartshorne. The problem states: For every degree $d>0$, and every $p=0$ or a prime number, give the equation of a nonsingular curve of degree $d$ in $\mathbb{P^{2}}$ over a field of…
Link
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An explanation of the arithmetic use of models of Shimura varieties

In the articles I browse that talk about the construction of local models of Shimura varieties, I often see them say things like "These models are useful arithemtically for the calculation of the semi-simple zeta function by means of the Lefchetz…
Tom Lewia
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