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It is a general fact that on any simple abelian variety, an effective divisor is ample. The following result underlies the usual algebraic proof of the projectivity of abelian varieties (defined initi …

A well-known theorem of Grauert and Fischer states that a smooth proper family of complex manifolds is a locally trivial fibration as soon as all the fibers are isomorphic. It is also easy to obtain a …

The discriminant locus has the following geometric interpretation, given in the introductory chapter of [Gelfand, Kapranov, Zelevinsky: Discriminants, Resultants and Multidimensional Determinants]. L …

All such resultants and/or discriminants are geometrically irreducible in characteristic zero, and a power of an irreducible in general. This is actually covered by the geometric argument I quoted in …

On an abelian variety (regardless of the characteristic), an effective divisor with positive self-intersection is ample. To be more precise, it suffices here to recall that on any simple abelian varie …

You can show that a smooth quadric in $\mathbb{P}^3$ is GL-equivalent to the product $\mathbb{P}^1 \times \mathbb{P}^1$ embedded by the Segre map. Once you know this, you're done. (A curve on $\mathb …

Thank you very much for this answer! This is not going to fit in a comment, so I'm writing another answer. [EDIT: I had hastily written some incorrect statements, now corrected in the text below. M …

abx's comment was made while I was writing this, but I am posting it as an answer anyway. There has not been a proof of this conjecture of Lang, which remains a wide open problem. Lu and Miyaoka's pa …

I have this basic question on which, strangely enough, the algebraic dynamics literature appears to be silent. But the question does not appear to be totally trivial or uninteresting to me - am I wron …

Yes! (I assume it was implicit in your question that the variety be projective?) More generally: any smooth, complex, rationally connected projective variety is simply connected. See Debarre's book …

Will Sawin and Michael Stoll have noted that, as a consequence of Faltings's "Big Theorem," a hyperelliptic equation $y^2 = f(x)$ with $\deg{f} > 6$ (genus $> 2$) and not admitting a degree $2$ non-co …

In Colmez's formulation, it is necessary that the endomorphism ring be the maximal order $\mathcal{O}_k$. It is then proved only in special cases ($k/\mathbb{Q}$ abelian), or on average over the CM ty …

Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg_LX = c_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $ …

In Arakelov geometry, the conventional wisdom is that the ``closed fibre at $\infty$'' should be viewed as totally degenerate. This is the extreme opposite of smoothness. A visualization in the case o …

The classical de Franchis theorem, as generalized by S. Kobayashi and T. Ochiai ("Meromorphic mappings onto compact complex spaces of general type," Inventiones, 1975), states that if $X$ is a complex …