13 votes

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 ...
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  • 726
12 votes

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-...
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  • 13.1k
12 votes
Accepted

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 ...
12 votes
Accepted

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 ...
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  • 6,712
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 ...
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  • 117k
9 votes

Strict transform of a tangent curve under blow-up

If $C \subset X$ is a smooth curve and $p \in C$ there is a unique plane (so-called osculating plane) $$ T^2_pC \subset T_pX $$ in the tangent space $T_pX$ to $X$ at $p$ such that $C$ any element of $...
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  • 31.9k
9 votes
Accepted

Open complement of hypersurfaces

The answer is no. Perhaps the simplest case is $n=2$, $d=4$. There is a unique double covering $\pi _i:S_i\rightarrow \mathbb{P}^2$ branched along $H_i$. If $U_1$ and $U_2$ are isomorphic, $S_1$ ...
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  • 34.7k
8 votes

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

Yes there is such a general statement, except for the fact that (as Nicolas suggests in his answer) you need to consider sequences of blow-ups with centers at different subspaces, not just points. The ...
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8 votes
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Finite group action on quasi-projective varieties

Let $X$ be a normal variety over an algebraically closed field of characteristic 0 with a finite group $G$ acting effectively. Since $G$ is finite it is reductive and a geometric quotient $X/G$ ...
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  • 8,044
8 votes
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Compact hyperkahler manifold as algebraic variety in weighted projective space?

If $X$ is a hypersurface and $\dim(X) > 2$ then by Lefschetz hyperplane theorem $H^{2,0}(X) = 0$, hence $X$ can't by hyperkahler.
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  • 31.9k
8 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 $$\operatorname{Pic}(\mathbb{P}^n) \to \operatorname{Pic}(X)$$ ...
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8 votes
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Geometry of critical points of holomorphic maps in projective space

For $n=2$, the locus $J$ is smooth and irreducible for a general $f$; i.e., these $f$ form a Zariski dense subset of the parameter space of such $f$. For $n\ge3$ and for general $f$, the locus $J$ ...
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7 votes
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A non-rational variety with a full exceptional collection?

Rationality of a variety with a full exceptional collection is a well-know folklore conjecture. In some form a similar open question is mentioned in the paper of Brown and Shipman "The McKay ...
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  • 31.9k
7 votes

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

$(P^1)^n$ has Poincaré polynomial $(1+t^2)^n$, and $P^n$ has Poincaré polynomial $1+t^2+ \dots + t^{2n}$. If $X$ is of dimension $n$ and $B_pX$ the blow-up at a point $p \in X$ then $p_{B_pX}(t) = p_X(...
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7 votes
Accepted

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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  • 4,745
7 votes

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 ...
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  • 31.9k
7 votes
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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 ...
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  • 31.9k
6 votes

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 ...
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  • 2,859
6 votes

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 ...
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  • 857
6 votes
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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 ...
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  • 4,442
6 votes
Accepted

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 ...
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  • 6,468
6 votes

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 ...
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6 votes

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 &...
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  • 726
6 votes

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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6 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 ...
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5 votes

How many holes may a projection of an algebraic variety have?

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})...
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  • 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 ...
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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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  • 41.6k
4 votes
Accepted

When is a monomial rational map on the projective space birational?

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 ...
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