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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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Relation between blowing up at a point and at a variety

maybe this is an idiot question, but I have to ask it. Usually, in the classical background, one defines the blow up $Bl_Y(X)$ at a variety $Y$ in as the closure of the graph of the function $f: X/Z(f_1, f_2, ...f_m) \longrightarrow…
user40276
  • 5,283
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2 answers

Pullbacks and pushouts

Say you have a curve $X$ of genus $g \geq 2$, and a surjective map $\phi:X \to E$, where $E$ is an elliptic curve. Denote by $J_X$, $J_E$ the Jacobians of $X$, $E$ respectively. Then we get an induced pushout map ${\phi}_\ast : J_X \to J_E$ and an…
anon1234
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Help in a question of Hartshorne's algebraic geometry book

I'm trying to solve this question: Let $\psi:\mathbb P^1\times \mathbb P^1\to \mathbb P^3$ be the Segre embedding given by $\psi([a_0:a_1],[b_0:b_1])\to [a_0b_0:a_0b_1:a_1b_0:a_1b_1]$. This corresponding to the ring homomorphism…
user42912
  • 23,582
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Why this property holds in a Veronese surface

I'm trying to understand this property of the Veronese surfaces which is an exercise in Hartshorne's book as well: Question: Let $Y$ be the image of the $2$-uple embedding of $\mathbf P^2$ in $\mathbf P^5$. This is the Veronese surface. If…
user42912
  • 23,582
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Which points of a cubic are the single intersection between that cubic and a conic?

I'm having problems with this exercise: For which points $P\in C$ ($C$ is a non-singular cubic) there exists a non-singular conic $Q$ with $C\cap Q = P$? Maybe we can use Pappus' theorem or some other application of Noether's theorem? Any ideas?
Mathematics
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Is a rational map in which fibers are generically singletons necessarily birational?

This is a very basic question about algebraic geometry which for some reason I am having trouble thinking about clearly. Let $X,Y$ be irreducible varieties over an algebraically closed field $k$ (in case this is ambiguous, definition below). Let $f$…
Ben Blum-Smith
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What is the linear space of the projective space $\mathbb{P}^n$

I ran into an exercise saying that Any projective variety is isomorphic to the intersection of a Veronese variety with a linear space. But I don't understand the definition of the term "linear space" in a projective space. In fact, I don't even…
hxhxhx88
  • 5,257
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1 answer

Twists and adjunction formula

Let $\pi:X\to S$ be a morphism of relative dimension one, where $X$ is a regular surface over a regular scheme $S$ of dimension one (everything over $\mathbb C$). We denote by $\omega_\pi$ the relative dualizing sheaf $\omega_X-\pi^\ast \omega_S$.…
Brenin
  • 14,072
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Miranda's book lemma 4.12 pag. 159

Suppose that the homogeneous coordinates of $\mathbb{P}^{n}$ are $[x_0:\dots:x_n]$ and that $H$ is defined by the linear equation $L=\sum_i a_ix_i=0$. Let the holomorphic map $\phi$ be defined by $\phi=[f_0:\dots:f_n]$ and set $D=-\min_i…
TheWanderer
  • 5,126
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1 answer

Does every open affine subscheme of an affine scheme have form $A_f$

Let Spec${A}$ be an affine scheme, can every open affine subscheme be written as Spec$A_f$ for some $f$ in $A$?
user93417
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Dimension and irreducibility of a family of curves in projective space

My problem concerns the construction of Mumford's famous example of an Hilbert scheme that has a non-reduced component. Let $k$ be an algebraically closed field, and let $X\subset \mathbf{P}^{3}_{k}$ be a regular cubic surface. Let $H$ be the…
Fq00
  • 1,221
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Question about Hartshorne problem (II 4.7)

I am in a weird situation with problem II 4.7 in Hartshorne's Algebraic Geometry. I can do parts b, c,d and e and I understand what to do in part a. but I am stuck on technical details. Here is the set up. We have a variety over $ \mathbb{C}$ and a…
DBr
  • 4,780
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1 answer

Embedding $\mathbb A^2-(0,0)$ into $k^n$, the image is not closed

For an arbitrary embedding $\mathbb A^2-(0,0)$ into $k^n$, must the image be not closed? In Mumford's Redbook p25, he argued as every coordinate function can be extended to $\mathbb A^2$, so the morphism can be extended to $\mathbb A^2$, so the…
user93417
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0 answers

Hilbert (sub)scheme: $\eta\in \underline{\textrm{Hilb}}^X_P(T)\,\,\,\iff\,\,\,?$

Let $H=\textrm{Hilb}^r_P$ be the Hilbert scheme of closed subschemes of $\mathbb P^r$ with fixed Hilbert polynomial $P$. Let $X\subset \mathbb P^r$ be any subscheme. I am trying to characterize $\textrm{Hilb}^X_P$ as a subscheme of $H$. So I take…
Brenin
  • 14,072
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1 answer

transversal intersection Hartshorne Lemma V.1.2 p.358

I try to understand an argument on page 358 of Hartshorne, proof of Lemma V.1.2. Consider a surface X, irreducible curves $C_i $on X and a very ample divisor D. Hartshorne says that since the intersections $C_i \cap D$ are nonsingular, the $C_i$ and…
claudi
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