4
$\begingroup$

This question is inspired by abx's comment to my previous question MO430933.

Let $X$ be a complex surface of general type, and denote by $$a \colon X \to \operatorname{Alb}(X)$$ the Albanese map of $X$.

It is classically known that the image $a(X)$ can be singular: for instance, if $C$ is a hyperelliptic curve of genus $3$ and $X=\operatorname{Sym}^2(C)$, then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is birational and contracts the $(-2)$-curve $\Gamma \subset X$ corresponding to the hyperelliptic involution of $C$ to an ordinary double point of $a(X)$.

However, in this example, the presence of the rational curve $\Gamma$ on $X$ implies that $\Omega_X$ is not globally generated. Moreover, this kind of situation is the only one I know in which there are singularities in the Albanese image. So, let me ask the

Question: What is an example of surface $X$ such that $\Omega_X$ is globally generated and $a(X)$ is singular?

Note that the global generation of $\Omega_X$ implies that the Albanese map is unramified, namely, a local immersion at each point $x \in X$.

Improve this question
$\endgroup$
2
  • $\begingroup$ Probably the surface parameterizing lines on a cubic threefold with an ordinary double point works. I believe the analysis of such specializations is how Clemens-Griffiths originally proved some of their results about the Abel-Jacobi map for cubic threefolds. The specialization was analyzed more carefully in work of Collino . . . $\endgroup$
    – Jason Starr
    Commented Sep 27, 2022 at 23:35
  • $\begingroup$ @JasonStarr: thank you very much! I will have a look at this example. $\endgroup$
    – Francesco Polizzi
    Commented Sep 28, 2022 at 5:22

0

Reset to default

You must log in to answer this question.