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164 views

Exercise in David Cox “Toric Varieties”

I want to do an exercise in the book Toric Varieties (by David Cox) Exercise 3.3.5. Let $\overline{\phi}:N \rightarrow N'$ be a surjective $\mathbb{Z}$-linear mapping and let $\widehat{\sigma}$ ...
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1answer
292 views

How to find the canonical divisor on a nonsingular toric variety?

I am reading Fulton's "Toric Varieties." In it, he explains that if $X$ is a toric variety and if $D_1, \ldots, D_d$ are the irreducible divisors invariant under the big torus action, then $$ ...
3
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1answer
196 views

Explicit example of a toric flip

I am looking for a toy example of a flip between toric projective 3-folds. More precisely, I would like to see their defining fans (or polytopes). Does anyone know where I can find something like ...
9
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1answer
266 views

Hodge theory for toric varieties

Say we are given a complex smooth projective toric variety $X$. How can one read off hodge theoretic information from combinatorial data? For example I would like to extract dimensions of the various ...
2
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1answer
100 views

differential with logarithmic poles

where can I find the computation of the groups $H^i(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^j)$? Moreover, if $D$ is a divisor with normal crossing in $\mathbb{P}^n$, how can I compute the hypercohomology ...
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1answer
94 views

figure out a simple toric variety

consider the plane cones $s=\langle(1,0),(1,n)\rangle$ and $t=\langle(1,0),(1,-n)\rangle$. This produce a toric variety obtained glueing $k[x,xy^n]$ and $k[x,xy^{-n}]$ along $k[x]$. There is a more ...
4
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1answer
296 views

Cohomology of $\mathcal O_X$ for toric varieties

Motivated by my ignorance here, if $X$ is a projective toric variety, is $$H^m(X, \mathcal O_X) \cong \begin{cases} 0 & m > 0 \\ \mathbb C & m = 1 \end{cases} $$ as for $\mathbb ...
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383 views

Equations of a projective toric variety

Given complete fan $\Delta$ defining a projective toric variety (so that $\Delta$ is the normal fan of some polytope). How do one go on to find a defining ideal of the toric variety in projective ...