Timeline for Which algebraic varieties admit a morphism to a curve?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Sep 22, 2017 at 21:47 | comment | added | R. van Dobben de Bruyn | Your argument that having a map to a genus $g \geq 1$ curve is a birational invariant is wrong. Not every morphism $X \to C$ factors through $\operatorname{Alb}_X$ (take $X = C$!). However, the result is still true: if $X'$ is birational to $X$, you get a rational map $X' \dashrightarrow C \to \operatorname{Jac}_C$, which by Weil's extension theorem extends to an actual map. It lands in $C$ because it does so on a dense open. | |
| Aug 14, 2010 at 0:34 | comment | added | damiano | @Dmitri, you are right. I will not edit the answer to reflect this, since the correction is already contained in the comments here and in jvp's answer. Nevertheless, the implication needed for the statements about the Neron-Severi group holds, and this is indeed obvious! | |
| Aug 13, 2010 at 22:16 | comment | added | Dmitri Panov | Donu, you are right, I did not read the answer of jvp, before commenting | |
| Aug 13, 2010 at 22:02 | comment | added | Donu Arapura | It appears that jvp discussed a version of this counterexample together with a (nontrivial) correction below. | |
| Aug 13, 2010 at 21:55 | comment | added | Dmitri Panov | The following statement is wrong: "An equivalent formulation of the question is the following: does X admit two disjoint effective non-zero nef divisor". Indeed, consider and Abelian surface A and take a generic non-trivial bundle L with c_1(L)=0. Then consider the projectivisation of L+O over A (O is the structure sheaf). Obviously this P^1 bundle has two disjoint sections that are effective and nef. It is easy to see that if A does not admit a map to an elliptic curve, then P(L+O)(A) does not admit a map to CP^1, contrary to what you are claiming. | |
| Aug 13, 2010 at 11:55 | comment | added | Angelo | The theorem of Castelnuovo and de Franchis has been generalized to arbitrary dimension by Catanese, Moduli and classification of irregular Kähler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math. 104 (1991),263-289. | |
| Aug 13, 2010 at 10:42 | history | edited | damiano | CC BY-SA 2.5 |
added a missing case
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| Aug 13, 2010 at 8:43 | history | answered | damiano | CC BY-SA 2.5 |