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In algebraic geometry, a projective variety over an algebraically closed field $k$ is a subset of some projective $n$-space $\mathbb P^n$ over $k$ that is the zero-locus of some finite family of homogeneous polynomials of $n + 1$ variables with coefficients in $k$, that generate a prime ideal, the defining ideal of the variety

6 votes

CY fibration over $\mathbb P^1$ without any singular fibers

To complement YangMills's answer: Viehweg and Zuo ("On the isotriviality of families of projective manifolds over curves") proved the following: Theorem. Let $X$ be a complex projective manifold of …
Piotr Achinger's user avatar
2 votes

The different gradings of a graded ring, and their schemes

EDIT. I misread the question - below I'm trying to describe the set of all gradings and not of their ${\rm Proj}$. Note that giving a $\mathbf{Z}$-grading on a $k$-algebra $A$ is equivalent to giving …
Piotr Achinger's user avatar