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By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ (h^{1 …

Let $X$ be a smooth scheme of finite type over $\mathbb Q$ or $\mathbb Z$. It is natural to consider $D(X)$, the derived category of coherent sheaves on $X$. Beyond the definition, have there been any …

It is well-known that the compactification $\overline M_{1,1}$ of the moduli space of elliptic curves over $\mathbb C$ is a weighted projective line with weights $4$ and $6$. As far as I can tell, th …

Let $X$ be a curve embedded into a projective space $\mathbb P$ such that it is cut out (scheme-theoretically or ideal-theoretically) by quadrics. What is known about the Koszul dual of the homogene …

This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that …

In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the quo …

Has anyone come across anything along the following lines? Let $X_1(p)$ be the compactification of the quotient of upper half plane by $\Gamma_1(p)$ for some unspecified large prime. Let $X_1(p) \to …

Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it being a member of a base-point-free linear system in a nef-Fano fourfold? What, in anything, is known rega …

I assume that you mean $x_1=x_2=0$ in your description of the divisors. I believe that you only really get two different resolutions, but yes, they are both projective varieties. Hodge numbers of bira …

In my paper arXiv:math/9802052 I give an argument in the graded case. The general case can be approached similarly (as sketched in the paper). The sequence in question is made from log-derivatives of …

I suspect that you will have trouble with trying to "simultaneously" resolve $X$ and $X/G$. We actually run into this issue in the joint paper with Anatoly Libgober arXiv:math/0206241 (although it may …

Even if you just have a fibration with one sphere as the base and the other as the fiber, you will have Euler characteristics $4$ which is not the Euler characteristics of $G(2,5)$. The description o …

The first requirement is that the toric variety is $\mathbb Q$-Gorenstein, otherwise discrepancies and crepancy are not defined. Combinatorially, this means that for every cone of the fan $\Sigma$ the …

There seems to be some terminology drift here. I would say that "order" would be called degree in modern terminology, for example. Here is the way I see it, and please someone correct me if I am wron …

Let me address the concrete problem. If you have two forms like this, the difference would be holomorphic, thus a constant, so uniqueness is clear. For existence, suppose the points are $a$, $b$ and …