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The following question seems very intuitive, but I haven't been able to find any proof (or counterexample). Let $X$ be a non-singular projective variety of $\dim X\ge 2$ and let $NS^1(X)$ be its …

Even though there are several examples showing that the Integral Hodge conjecture (IHC) fails in general, there are some positive results as long as the canonical bundle is not too positive: Theorem …

Yes. The Segre embedding $i:P:=\mathbb P^n \times \mathbb P^m\to \mathbb P^N$ is defined by the sections of the line bundle $O_P(1,1):=pr_1^*O_{\mathbb P^n}(1)\otimes pr_2 O_{\mathbb P^m}(1)$ on $\mat …

I think you want this article by J. Sidman and P. Vermeire.

Weighted projective spaces $\mathbb{P}(a_1,\ldots,a_n)$ are examples where a grading other than the standard grading is used. In general you can study gradings coming from any finitely generated abeli …

Your question appears as the 'Complement problem' in Hanspeter Kraft's article Challenging problems on affine n-space and it seems from that article that the problem is still open for $H_1, H_2$ irred …

Your problem basically boils down to computing the groups $H^i_Z(X,\mathcal O)$, the local cohomology with support in $Z$. Once these groups are known the local cohomology exact sequence takes the fo …

Here is a way of generating lots of examples: Start with a variety $X$ with an effective cone which is not rational polyhedral (i.e., not finitely generated) and let $L_1,\ldots,L_r$ be a collection …

Write for simplicity $X=\mathbb{P}^n$. The easiest way of showing 1) is probably by noting that $\mathcal I / \mathcal I^2$ injects as a subbundle of $\mathcal O_Y(-1)^{n+1}$ (this follows from comb …

No. There is a lot of research focusing on singular rational plane curves and there are many open problems here. For example, it is not known how many cusps a rational irreducible plane curve can have …

I don't know if there is an elementary proof out there which works for all hypersurfaces of degree $d\ge n$, but if $X$ is general, you can see quite easily that they contain projective lines - or equ …

Yes, $L$ is an ample line bundle and hence $\Sigma_k$ is a smooth ample divisor. The Lefschetz hyperplane theorem now implies that $\pi_1(\Sigma_k)\to\pi_1(X)$ is surjective.

This is unlikely unless $X$ is say, a hypersurface. The reason is that the Euler characteristic is invariant of the projective embedding (it is a topological invariant), while the equations defining a …

The sections $s_1=x^3,s_2=x^2y,s_3=xy^2$ and $t_1=x^2,t_2=xy,t_3=y^2$ are not the same sections, in fact they correspond to different line bundles. The $s_i$ do have a base-point, and as you remark, …

I think the following construction could work: Let $X$ be a Calabi-Yau threefold without any rational curves (such threefolds do exist). Then $Nef(X)=\overline{Eff}(X)$, i.e., the nef cone equals the …