# Tag Info

### Blow-up of projective variety $P^1 \times P^1..... \times P^1$ ($n$ times) and blow-up of $P^n$

The 'permutohedral variety' (the toric variety obtained from the permutohedron) is a very attractive toric variety which can be obtained by blowing up all of the toric strata in $\mathbb P^n$ in ...
• 726

### Moduli space of flat connections over a Riemann surface

Both of these moduli spaces are discussed in the survey "Flat connections on oriented 2-manifolds" by Lisa Jeffrey. The theme of the first section of the paper is roughly as follows. An oriented 2-...
• 13.1k
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### Recover the characteristic of $k$ from the category of $k$-varieties

Correction. As correctly noted by Remy van Dobben de Bruyn, there is a mistake in Lemma 4. What follows is a corrected argument, with the original (mistaken) post appended below the corrected ...
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### Degree of secant varieties of Veronese varieties

The secant variety $Sec_k(V^n_2)$ is the variety parametrizing $(n+1)\times (n+1)$ symmetric matrices modulo scalar of rank at most $k$ that is of corank at least $n+1-k$. Then by Proposition 12(b) in ...
• 6,712

### 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 ...
• 117k

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### Linear spaces secant to Veronese varieties

Here is an answer in terms of power sum decompositions of polynomials. A point $p \in \mathbb{P}^9$ corresponds to a homogeneous polynomial $P$ of degree $3$ in $3$ variable, defining a plane cubic. ...
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### Open complement of hypersurfaces

The easiest case is $n = 1$, $d = 4$. Indeed, the embeddings $U_i \to \mathbb{P}^1$ are canonical, hence an isomorphism $U_1 \cong U_2$ extends to an isomorphism of the ambient projective lines and ...
• 31.9k
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### Square root of a line bundle up to a finite surjective morphism

Assume $\mathcal{L}$ is associated with an effective Cartier divisor $D$. Let $D'$ be another Cartier divisor such that $D + D'$ is divisible by 2 in $\mathrm{Pic}(X)$. Let $$g \colon X' \to X$$ be ...
• 31.9k

### Blow-up of projective variety $P^1 \times P^1..... \times P^1$ ($n$ times) and blow-up of $P^n$

[This is a correction of an earlier answer, and is currently incomplete; thanks to Nicolas for pointing out that what I had written was totally wrong.] Both varieties are rational. The obvious ...
• 2,859

### existence of birational morphism and divisors

I don't know where to find a proof written, but it is not hard to give one here. One direction is easy. If $S \rightarrow \mathbf P^2$ is a birational morphism, then let $D$ be the pullback of a ...
• 857
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### Irreducible components: associativity for intersections?

No. This is an example with irreducible (and nonsingular) $A,B,C$: Consider $\mathbb{A}^4$ with coordinates $(w,x, y, z)$. Let $B$ be the $(x,y)$-plane (i.e. the set $w = z = 0$), $C$ be the ...
• 4,442
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### CY fibration over $\mathbb P^1$ without any singular fibers

No such variety exists: first, you can use Remark 3.2 here to see that your map $\pi$ must be a holomorphic fiber bundle, and then Lemma 17 here gives you that this bundle becomes a trivial product ...
• 6,468

### CY fibration over $\mathbb P^1$ without any singular fibers

To complement YangMills's answer: Viehweg and Zuo ("On the isotriviality of families of projective manifolds over curves") proved the following: Theorem. Let $X$ be a complex projective ...
• 13.8k

### Smooth complete intersections

If $n=5$ then let $\mathbb P^5$ have coordinates $x_0,\ldots,x_5$ and suppose the plane is $H=\mathbb P^2_{(x_0:x_1:x_2)}$. The two equations of $X$ are necessarily of the form  \begin{pmatrix} A_1 &...
• 726

### Geometry of critical points of holomorphic maps in projective space

For n=1, every point in the critical divisor has degree $≤d-1$, where $d$ is the degree of the map, and the total degree of the critical divisor is $2d−2$, and any such divisor can occur. To state it ...
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### 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 ...
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Blow up to get a morphism $\Pi: Bl_{P_0}\mathbf P^n \rightarrow \mathbf P^{n-1}$. Let $\widetilde{V}$ be the proper transform of $V$ in $Bl_{P_0}\mathbf P^n$. Then $\overline{\pi(V)}=\Pi(\widetilde{V})... • 857 5 votes ### Unsplitting sequence of vector bundles$\def\CC{\mathbb{C}}$A splitting would be a global map$f : G(k,n) \times \CC^n \to \CC^n$such that$f(L,v) \in L$for all$L \in G(k,n)$and$v \in \CC^n$. But, since$G(k,n)$is projective and ... • 140k 5 votes ### Are "transverse" hyperplane sections of nondegenerate irreducible projectice varieties always nondegenerate This is a bit of a folk theorem. Harris (Algebraic Geometry, Proposition 18.10 and Exercise 18.11) states it for general hyperplane sections, but actually proves it for all generically transverse ... 4 votes Accepted ### A sub-variety of a Grassmannian Per my very recent answer on another question (https://mathoverflow.net/a/266282/66), consider an invertible linear transformation with two eigenspaces, both of dimension$n$(for example,$\mathrm{...
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There is no characteristic restriction needed. Here's a direct proof for $n=2$ (below the general case is proved based on elementary linear algebra). I write coordinates $(x:y:z)$. Since your ...