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In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then t …

This is false, even in dimension $3$. Take a regular icosahedron. Then wiggle the vertices a bit -- still get an icosahedron, but it is not regular. If you take the barycenters and the convex hull, yo …

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 …

Sorry, don't know a reference, but here is a quick argument. If $M=p^ab+c$ with $0\leq c\leq p^a-1$, then $$(1+x)^M=(1+x)^{p^ab}(1+x)^c =(1+x^{p^a})^b(1+x)^c \mod p. $$ In turn, this equals $$ (1+bx …

The solution set $L$ is clearly a sublattice in $\mathbb Z^d$. If $a$ is fixed and $N$ grows, the number of lattice elements in a box grows approximately as $cN^{rank}$. So if you have your statement …

There is a toric complete intersection example, rather ad hoc, in arXiv:alg-geom/9402002. This particular example is related to the situation when a decomposition of anticanonical class on toric Fano …

It is not really my area, but the answer to part 2 is that you are looking for the smallest $N$ such that there are exactly $8n-4$ ways of writing it in the form $a^2+b^2$. If such $N$ is even but i …

I would like to add that (cohomology of) chiral de Rham complex has to be viewed as the large Kahler limit of halftwisted theory. Indeed, it does not use the Kahler data on the CY manifold and does no …

If you are willing to work with smooth stacks, rather than smooth toric varieties (or alternatively consider toric varieties with quotient singularities) then it is definitely possible. Combinatoria …