Obstructions to being a hyperplane section or a fibre of a Lefschetz pencil - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:23:31Z http://mathoverflow.net/feeds/question/80027 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80027/obstructions-to-being-a-hyperplane-section-or-a-fibre-of-a-lefschetz-pencil Obstructions to being a hyperplane section or a fibre of a Lefschetz pencil A. Pascal 2011-11-04T09:35:10Z 2011-11-04T14:31:35Z <p>Given a smooth projective variety $X$, when could $X$ fail to be a hyperplane section in some other variety $Y$, or fail to be the fibre of some Lefschetz pencil $\widetilde{Y} \rightarrow \mathbb{P}^{1}$? </p> <p>Here, the variety $Y$ is not fixed, but simply required to exist.</p> http://mathoverflow.net/questions/80027/obstructions-to-being-a-hyperplane-section-or-a-fibre-of-a-lefschetz-pencil/80037#80037 Answer by jvp for Obstructions to being a hyperplane section or a fibre of a Lefschetz pencil jvp 2011-11-04T11:28:44Z 2011-11-04T11:35:07Z <p>Your question is related to the problem of the existence of non-trivial extensions of subvarieties $X \subset \mathbb P^N$ to $\mathbb P^{N+1}$. An extension of $X$ is just a subvariety $Y$ of $\mathbb P^{N+1}$ such that $X= Y \cap \mathbb P^{N}$. It is called trivial if $Y$ is the join of $X$ and a point outside of $\mathbb P^N$.</p> <p>This is a classical question that was studied by the Italian school of algebraic geometry. For instance, Scorza proved that the Veronese surface in $\mathbb P^5$ does not admit non-trivial extensions. </p> <p>More recently, the problem has been studied by Zak, S. L'vovsky, L. Badescu, among many others. In <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.mmj/1029004452" rel="nofollow">Extensions of projective varieties and deformations</a> by S. L'vovsky you will find the following result:</p> <p><strong>Theorem.</strong> Suppose $X$ is not $\mathbb P^N$ nor a quadric. If $\dim X \ge 2$ and $H^1(X,TX\otimes \mathcal O_{\mathbb P^N}(-1))=0$ then every extension of $X$ is trivial. </p> <p>For a very nice introduction to this circle of ideas see the first chapter of the book <a href="http://books.google.com/books/about/Projective_geometry_and_formal_geometry.html?id=WtsHOfktVBsC" rel="nofollow">Projective geometry and formal geometry</a> by L. Badescu. Unfortunately, the relevant Chapter doesn't seem to be available from Google books.</p> <p>Of course this does not answer your question as the embedding of $X$ into $\mathbb P^N$ is fixed. </p> http://mathoverflow.net/questions/80027/obstructions-to-being-a-hyperplane-section-or-a-fibre-of-a-lefschetz-pencil/80048#80048 Answer by ulrich for Obstructions to being a hyperplane section or a fibre of a Lefschetz pencil ulrich 2011-11-04T13:06:20Z 2011-11-04T14:31:35Z <p>A generic curve over $\mathbb{C}$ of large genus, say at least $24$, will not be a fibre of a Lefschetz pencil or even a hyperplane section.</p> <p>The reason is that by the theorems of Harris-Mumford and Eisenbud-Harris, the moduli space of curves of large genus is of general type, so there can be no rational curve passing through a generic point. To use this one needs to know that all the smooth fibres of the pencil are not isomorphic. Since the local monodromy around a singular fibre is infinite by the Picard-Lefschetz formula, it suffices to show that there must be at least one singular fibre. But if all fibres are smooth, then $\tilde{Y}$ (the total space of the Lefschetz pencil) must be isomorphic to $C \times \mathbb{P}^1$. This cannot happen since $C \times \mathbb{P}^1$ is not a blow up of any other surface.</p> <p>Over a field of characteristic zero any very ample linear system contains a Lefschetz pencil, so it follows from the above that the smooth members of the linear system cannot all be isomorphic. But this would give rise to a unirational variety containing a generic point of the moduli space and this is not possible.</p> <p>A similar argument should work for other classes of varieties whose moduli spaces are not uniruled.</p>