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I assume you mean dim 3 ODPs? The $A_1$ singularity typically refers to surface ODPs. I don't think such variety can be constructed. On the fan side, the minimum generators of the rays form somethin …

Is there a reference for generators of fundamental groups of (some) fake projective planes in terms of matrices in $SU(2,1)$?

If $X$ and $Y$ are smooth and proper, then this looks a lot like the $K$-equivalence, which states that for some (=any) smooth proper $Z$ that maps to both $X$ and $Y$ the exceptional divisors are the …

No reason for that. If $L$ and $L_0$ are generic (given their degrees), then the equality of dimensions will hold. But there is no reason to have a natural isomorphism of spaces of sections.

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 …

Let $X$ be a smooth projective curve over $\mathbb C$. Let $D$ be a divisor on it. What is known about upper bound on dimension of the cokernel of $$Sym^2(H^0(X,D)) \to H^0(X,2D)?$$ In my case the div …

Can anyone please provide me with a reference on $H^n(M_{0,n+3},{\mathbb C})$ where $M_{0,n+3}$ is the (affine) scheme parametrizing $n+3$ labeled distinct points on ${\mathbb C\mathbb P}^1$? I am loo …

You can think of this ring as the semigroup ring of the semigroup $S$ generated by $$(1,0,0,1),(-1,0,0,1),(0,1,0,1),(0,-1,0,1),(0,0,1,1),(0,0,-1,1).$$ The above semigroup elements correspond to $f_1,f …

I have an explicit genus one curve $E$ with two points $p_1$ and $p_2$ on it and am looking for an explicit degree seven cover $X\to E$ with ramification precisely over $p_i$, with a single preimage p …

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 …

There is an example of an octic with 84 A_2 singularities (known upper bound is 98) in arXiv:1108.1820, section 9. Also, check out this paper http://arxiv.org/abs/math/0505022

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 …

Assuming that the blowup in question is toric, and exceptional divisor has one component, you are not going to get anything beyond these examples. After all, it would mean that you have a subdivision …

An obvious thing to try is to consider the Newton polytope $\Delta$ of $f$ and take $\rm Proj$ of the corresponding semigroup algebra $$ P=\rm{Proj}\oplus_{k\geq 0}\mathbb C[k\Delta]. $$ Then $Z$ is $ …