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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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Understanding a lemma in proof of Valuative Criteria of Separatedness

I am reading this document here by Brian Osserman concerning the Valuative Criteria for properness and separatedness. In particular, I am trying to understand the second half of the proof Proposition 2.5. Here is the proposition: Proposition…
user38268
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Divisors, Sections, Vanishing

Let $D=\sum_{i=0}^n\left.n_iD_i\right.$ be a Weil divisor on a nonsingular, projective variety $X$. Then, it corresponds to a Cartier divisor $D$, which corresponds to a line bundle $\mathcal{L}:=\mathcal{L}(D)$ and there exists a section…
Jesko Hüttenhain
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Exercise on Complex Tori

I can't solve this exercise: Miranda Algebraic Curves and Riemann Surfaces pag. 43 K: Recall that a lattice $L \subset \mathbb{C}$ is an additive subgroup generated (over $\mathbb{Z}$) by two complex numbers $\omega_1$ and $\omega_2$ which are…
TheWanderer
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Naive algebraic geometry question

Please no flame, I am a beginner here... Given ANY subset of $\mathbb{C}$, is there always at least one set of polynomial equations which generates it? If yes, can we always explicitly construct at least one member of that set (of polynomial…
Frank
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Pullback of global sections

Let $i:X \hookrightarrow Y$ be a closed subscheme. Assume $X$ and $Y$ are projective schemes. Let $\mathcal{F}$ be an $\mathcal{O}_Y$-module. Under what condition on $\mathcal{F}$ or $X, Y$ can we conclude that the natural morphism from the global…
Chen
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Help in these really elementary definitions in Algebraic Geometry

Based on this great answer of my own post: Change of coordinates (algebraic variety) which the answerer really give me concrete examples of the $T$ of the second paragraph. I'm with the same kind of doubts in the first paragraph. I understood that…
user42912
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A simple question of blowing-up

In Harris' and Morrison's book "Moduli of Curves" page 123, they study stable reduction of cuspidal singularity appearing in the center fiber. When they first blow up the total family at the origin, why does the exceptional divisor $E_1$ have…
JacobI
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Kähler differentials of a hyperelliptic curve.

Let $f:X \rightarrow \mathbb{P}^1_k$ be a hyperelliptic curve, where k is a field of characteristic not 2. $\mathbb{P}^1_k$ is the union of $U = Spec k[t]$ and $V = Spec k[s]$ where $s= 1/t$. This gives us $f^{-1}(V) \cup f^{-1}(U) = X$. On pg. 292…
Tedar
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Why doesn't this argument prove every nonsingular curve embeds into the projective plane?

It is a statement that some curves can never be embedded in the plane, but only in projective 3-space. Why does the following argument not prove that every nonsingular curve can be embedded in the projective plane, though? Let $X$ be a nonsingular…
oggledog
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Euler character of a numerically trivial divisor

Let $X$ be a complex projective variety (might be singular) and $D$ be a Cartier divisor on $X$. Suppose $D$ is ${numerically}$ trivial, then is the Euler character $\chi(X,D)= \chi(X, \mathcal{O}_X)$? Here numerically trivial means the intersection…
Li Yutong
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ring of regular functions $\mathcal{O}(X)_f$ (after localization)

Let $X$ be an affine $k$-variety, and let $f$ be an element of $\mathcal{O}(X)$. The subvariety $D(f)$ of $X$ is a quasi-affine $k$-variety. Is $\mathcal{O}(X)_f$ the ring of regular functions of $D(f)$?
yannickvda
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does pull back very ample implies very ample?

Let $X$, $Y$ be proper schemes (can assume over a field) and let $L$ be a line bundle on $X$. Assume I have a finite morphism $q:Y\rightarrow X$ and that I know that $q^{*}L$ is very ample. Is it also true that $L$ is very ample? if not do you have…
user65187
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Finding $\operatorname{Aut}(X)$ where $X$ is the Fermat cubic surface in $\mathbb{P}^3$.

I am trying to understand the automorphism group of the Fermat Cubic surface $$x_0^3 + x_1^3 + x_2^3 + x_3^3 = 0$$ in $\mathbb{P}^3$ to solve Hartshorne's exercise V.4.16. The way I thought to go about this was to find a subgroup, and argue that the…
Daniel
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Ideal sheaves pullback via monoidal transform

Let $X$ be a surface with two divisors $D,E$. Suppose $D,E$ intersects at $P$, denote by $Z$ the scheme-theoretic intersection. Let $\mathcal{I}$ be the ideal sheaf corresponding to $Z$. Now we blow up $X$ at $P$, $\pi: \tilde{X} \to X$, with…
finiteness
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Can you graph $y^{4}x^{3}+x^{4}-y=0$ without a computer?

I'm trying to get an intuitive sense of algebraic varieties from their equations. Some varieties are more easily imagined without a computer since either $y$ or $x$ can be isolated, producing functions, and these can be more easily graphed. But what…
shintuku
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