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Results tagged with projective-varieties
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user 18060
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
3
votes
Accepted
Irreducible components of a projective variety
We have either $H(0,0,1)=0$ or $H(0,0,1) \neq 0$. In the first case, the locus $w=x_0=0$ is contained in $Z(H_1,\dots, H_n)$ and is either an irreducible component of dimension $n-1$ or contained in a …
- 148k
10
votes
Open complement of hypersurfaces
If $U_1$ and $U_2$ are isomorphic then $H_1$ and $H_2$ are equal in the Grothendieck ring of varieties and thus, by the Larsen-Lunts theorem, stably birational, which if $d>n$ implies that they are is …
- 148k
9
votes
Isomorphism of varieties in $\mathbb{C}$ implies isomorphism over finite fields
(I wrote this earlier and didn't finish posting it. It has some overlap with Remy's answer, but I figured it was worth posting to provide a slightly different perspective.)
No. For example the ellipti …
- 148k
5
votes
Accepted
A variation on the projective Nullstellensatz
Since $V(f_1,\dots,f_n)=0$, by the Nullestellensatz
$$ \sqrt{ (f_1,\dots, f_n)} = I ( V( f_1,\dots, f_n)) = I(0) = (x_1,\dots, x_r)$$
where $x_1,\dots, x_r$ are a basis for $V^*$.
So for each $i$ from …
- 148k
2
votes
Accepted
On relating $l(A), l(B)$ and $l(A+B)$ for Weil divisors on a smooth projective curve where o...
If $D$ and $E$ are linearly equivalent to effective divisors, this is OK from what's in Hartshorne, as both sides are invariant under linear equivalence.
If $E$, say, is not linearly equivalent to an …
- 148k
2
votes
Grothendieck rings and the Tannakian formalism
The conjectural category of motives, being an abelian category, has its own notion of $K$-theory, $K_0 ( \operatorname{Mot}_k)$.
This would be the quotient of the free abelian group generated by isomo …
- 148k
3
votes
Accepted
Moduli spaces of horizontal curves
It is not possible to have such a moduli space that contains all the smooth curves of genus $g$ and degree $d$ and over which the universal family of curves is proper.
Let $X = \mathbb P^1 \times \mat …
- 148k
3
votes
Accepted
Intersection in toric variety
If by "subvariety" you mean that $W$ is known to be irreducible, the answer is positive. Otherwise, it is negative, as I will show by an explicit counterexample.
Let $Z$ be the base locus of $D$. The …
- 148k