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Results tagged with projective-varieties
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user 7460
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
What is the relationship between determinantal varieties?
Set $M_k:=V(I_k) \subset \operatorname{Mat}(m \times n) \simeq \mathbb{A}^{mn}$.
(1) If $k < \min\{m, \, n\}$ then $M_{k-1}$ is precisely the singular locus of $M_k$. See
E. Arbarello, M. Cornalba, P …
- 66.3k
4
votes
Accepted
Varieties with disjoint prime divisors
Take a smooth curve $C$ of genus $\geq 2$ and consider a smooth divisor $D \subset C \times C$ which is $2$-divisible in $\operatorname{Pic}(C \times C)$ and which is transverse to both factors (by Be …
- 66.3k
5
votes
Varieties with few trisecant lines
You can have a look at Ingrid Bauer's paper
Bauer, I., The classification of surfaces in $\mathbb{P}^5$ having few trisecants, Rend. Semin. Mat., Torino 56, No. 1, 1-20 (1998). ZBL0965.14029.
It turns …
- 66.3k
7
votes
Smooth complete intersections
If $X \subset \mathbb{P}^n$ is a non-degenerate, smooth complete intersection variety of dimension at least $3$, then the restriction map $$\operatorname{Pic}(\mathbb{P}^n) \to \operatorname{Pic}(X)$$ …
- 66.3k