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As Damiano pointed out, if a smooth projective $X$ surjects onto a curve, and $\dim X>1$ then the Picard number is at least $2$. This gives a clear obstruction, which explains why $\mathbb{P}^n$ won't satisfy it (when $n>1$). Here is a small refinement:

Lemma: Suppose that an $n$ dimensional variety $X$ surjects onto a $m$-dimensional variety with $0<m<n$. Then $$\dim CH^m(X)\otimes im[CH^m(X)\otimes \mathbb{Q} \to H^{2m}(X)]>1$$ where $H^*(X)$ denotes singular or $\ell$-adic cohomology.

Proof: The class of the general fibre and $H^{n-m}$, with $H$ an ample divisor, are independent.

Cor: $\mathbb{P}^n$ does not surject onto a nontrivial lower dimensional variety.

(This can also be seen directly from Bezout's theorem.)

On the other hand, $X$ will map onto $\mathbb{P}^1$ after blowing up, as Charles observed, and this is often a very useful trick in practice.

The case of (*) $X$ mapping onto curves of genus two or more is actually something that has been studied a number of people*. From Castelnuovo-De Francis (see Damiano's answer) one can extract a number of topological criteria. Here's one: $X$ satisfies (*) if and only if the fundamental group admits a surjective homomorphism onto the fundamental group of such a curve. (In my original answer, I hadn't realized that JVP already discussed this.) Some of this probably described in the multi-author book on Kaehler groups. Also take a look at my note in the Bulletin from way back in the last century.

In case it wasn't clear the results in the last paragraph are characteristic $0$ only. The positive characteristic case has not been looked at seriously, as far as I know, and is potentially very interesting. Warning: the standard tecnhiques such as Castelnuouvo-De Franchis, will fail!

* Among them: Amoros, Beauville, Bressler, Campana, Catanese, Gromov, Green, Lazarsfeld, Ramachandran, Simpson Siu, and me. People coming from hyperplane arrangement theory have also looked at the question for open varieties, but here the literature is so large, I won't even attempt it.

The last edit was to rule out maps to points. I'm sure there is still some stupid exception that I'm overlooking.

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As Damiano pointed out, if a smooth projective $X$ surjects onto a curve, and $\dim X>1$ then the Picard number is at least $2$. This gives a clear obstruction, which explains why $\mathbb{P}^n$ won't satisfy it (when $n>1$). Here is a small refinement:

Lemma: Suppose that an $n$ dimensional variety $X$ surjects onto a $m$-dimensional variety with $m<n$0<m<n$. Then $$\dim CH^m(X)\otimes \mathbb{Q}>1$$ Proof: The class of the general fibre and$H^{n-m}$, with$H$an ample divisor, are independent. Cor:$\mathbb{P}^n$does not surject onto a nontrivial lower dimensional variety. (This can also be seen directly from Bezout's theorem.) On the other hand,$X$will map onto$\mathbb{P}^1$after blowing up, as Charles observed, and this is often a very useful trick in practice. The case of (*)$X$mapping onto curves of genus two or more is actually something that has been studied a number of people*. From Castelnuovo-De Francis (see Damiano's answer) one can extract a number of topological criteria. Here's one:$X$satisfies (*) if and only if the fundamental group admits a surjective homomorphism onto the fundamental group of such a curve. (In my original answer, I hadn't realized that JVP already discussed this.) Some of this probably described in the multi-author book on Kaehler groups. Also take a look at my note in the Bulletin from way back in the last century. In case it wasn't clear the results in the last paragraph are characteristic$0$only. The positive characteristic case has not been looked at seriously, as far as I know, and is potentially very interesting. Warning: the standard tecnhiques such as Castelnuouvo-De Franchis, will fail! * Among them: Amoros, Beauville, Bressler, Campana, Catanese, Gromov, Green, Lazarsfeld, Ramachandran, Simpson Siu, and me. People coming from hyperplane arrangement theory have also looked at the question for open varieties, but here the literature is so large, I won't even attempt it. The last edit was to rule out maps to points. I'm sure there is still some stupid exception that I'm overlooking. 3 added 4 characters in body As Damiano pointed out, if a smooth projective$X$surjects onto a curve, and$\dim X>1$then the Picard number is at least$2$. This gives a clear obstruction, which explains why$\mathbb{P}^n$won't satisfy it (when$n>1$). Here is a small refinement: Lemma: Suppose that an$n$dimensional variety$X$surjects onto a$m$-dimensional variety with $m<n$. Then $$\dim CH^m(X)\otimes \mathbb{Q}>1$$ Proof: The class of the general fibre and$H^m$, H^{n-m}$, with $H$ an ample divisor, are independent.

Cor: $\mathbb{P}^n$ does not surject onto a lower dimensional variety.

(This can also be seen directly from Bezout's theorem.)

On the other hand, $X$ will map onto $\mathbb{P}^1$ after blowing up, as Charles observed, and this is often a very useful trick in practice.

The case of (*) $X$ mapping onto curves of genus two or more is actually something that has been studied a number of people*. From Castelnuovo-De Francis (see Damiano's answer) one can extract a number of topological criteria. Here's one: $X$ satisfies (*) if and only if the fundamental group admits a surjective homomorphism onto the fundamental group of such a curve. (In my original answer, I hadn't realized that JVP already discussed this.) Some of this probably described in the multi-author book on Kaehler groups. Also take a look at my note in the Bulletin from way back in the last century.

In case it wasn't clear the results in the last paragraph are characteristic $0$ only. The positive characteristic case has not been looked at seriously, as far as I know, and is potentially very interesting. Warning: the standard tecnhiques such as Castelnuouvo-De Franchis, will fail!

* Among them: Amoros, Beauville, Bressler, Campana, Catanese, Gromov, Green, Lazarsfeld, Ramachandran, Simpson Siu, and me. People coming from hyperplane arrangement theory have also looked at the question for open varieties, but here the literature is so large, I won't even attempt it.

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