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Results tagged with projective-varieties
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user 108274
In algebraic geometry, a projective variety over an algebraically closed field $k$ is a subset of some projective $n$-space $\mathbb P^n$ over $k$ that is the zero-locus of some finite family of homogeneous polynomials of $n + 1$ variables with coefficients in $k$, that generate a prime ideal, the defining ideal of the variety
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1
answer
375
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Rational normal curve as determinantal variety
We work here over complex numbers. Let $\Omega(Z)$
\begin{pmatrix}
L_1 & L_2 & ... & L_n \\
M_1 & M_2 & ... & M_n\\
\end{pmatrix}
be a $2 \times n$ matrix of homogeneous linear forms
$L_i(Z), M_j(Z) …
- 6,018
2
votes
0
answers
215
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Clemens-Griffiths component birational invariant
Let $X$ be a smooth variety over complex numbers $\mathbb{C}$, say a threefold for sake of better intuition. Is there any geometrical intuition behind the fact that
the Clemens-Griffiths component of …
- 6,018
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290
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Chow's Lemma: Mumford's and Grothendieck's (?) definitions
David Mumford gives in his book Algebraic Geometry I, Complex Projective Varieties
on page 61 a definition of Chow's Lemma which has at least for me not a
usual form:
If says that a closed $^*$-analyt …
- 6,018
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0
answers
281
views
Pencil of divisors in algebraic geometry
Let $X \subset \mathbb{P}^n$ be projective variety over alg closed field of char $0$ and
$C = V(F), D= V(G) \subset X$ two distinct divisors (e.g. two quadrics,
curves or lines lying in a surface,...) …
- 6,018
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votes
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answers
125
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Quotient $(V -S)/G$ is a quasi-projective variety for every closed $S \subset V$ with free $...
I have a question about the statement of Remark 1.4 following on Thm 1.3 from Burt
Totaro's paper "The Chow Ring of a Classifying Space" (p. 4):
Let $G$ be a reductive group over a field $k$.
Remark …
- 6,018
2
votes
1
answer
350
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Comparison of classical and Zariski topologies with constructible sets
In David Mumford's book Algebraic Geometry I, Complex Projective Varieties the
proof of (3.25) Specialization principle on page 53 contains an argument
I not understand.
General assumptions: all our v …
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4
votes
1
answer
373
views
Universal hyperplane section and nondegeneracy of general hyperplane section
I have a question about Exercise 18.11 In Harris' book Algebraic Geometry, on page 231:
Give a proof of the nondegeneracy of the general hyperplane
section of an projective irreducible nondegenerated …
- 6,018
2
votes
1
answer
641
views
Unibranch points (definition for varieties over arbitrary field)
In David Mumford's book Algebraic Geometry I, Complex Projective Varieties
treating mainly complex varieties as objects of interest on page
43 he defines what is a topologically unibranch variety $X$ …
- 6,018
0
votes
1
answer
278
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Question about Correspondences from Mumford’s Complex Projective Varieties
I study David Mumford's Algebraic Geometry I - Complex Projective Varieties
and have some problems to understand a step in the proof of Lemma 6.7 (b).
Firstly, the general setting & preparations arou …
- 6,018
1
vote
1
answer
326
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Proj construction and nilpotent homogenous elements in graded ring
Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined map …
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