Picard group of a very ample divisor in a smooth variety of dimension >3 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:41:13Z http://mathoverflow.net/feeds/question/120131 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120131/picard-group-of-a-very-ample-divisor-in-a-smooth-variety-of-dimension-3 Picard group of a very ample divisor in a smooth variety of dimension >3 aglearner 2013-01-28T17:58:30Z 2013-01-29T15:44:40Z <p>Let $X$ be a smooth complex projective variety, $\dim(X)>3$ and $D$ be a very ample possibly reducible divisor on $X$. Is it true that $\textrm{Pic}(X)\cong \textrm{Pic}(D)$? If not, what would be a reasonable condition on $D$ (for example, would it suffice that $D$ is irreducible)? </p> <p>(if you have a precise reference, I would be especially grateful).</p> http://mathoverflow.net/questions/120131/picard-group-of-a-very-ample-divisor-in-a-smooth-variety-of-dimension-3/120135#120135 Answer by Francesco Polizzi for Picard group of a very ample divisor in a smooth variety of dimension >3 Francesco Polizzi 2013-01-28T18:13:59Z 2013-01-28T22:11:04Z <p>The answer is <strong>yes</strong> under very mild assumptions. In fact there is the following result.</p> <blockquote> <p><strong>Proposition.</strong> Let $L$ be a $k$-ample line bundle on a normal, irreducible, projective variety $X$ with at most Cohen-Macauley singularities. Assume that the dimension of the locus of non-rational singularities of $X$ is at most $0$ and that $D \in |L|$ is a divisor such that $X-D$ is a local complete intersection. Then, under the restriction mapping, we have $\textrm{Pic}(X) \cong \textrm{Pic}(D)$ if $\dim X \geq 4+k$ and $\textrm{Pic}(X) \to \textrm{Pic}(D)$ is injective with torsion free cokernel if $\dim X = 3+k$.</p> </blockquote> <p>The case you are interested in corresponds to $k=0$. For instance, if $X$ is smooth (of dimension at least $4$) then all the assumptions are fulfilled and what you want is true. </p> <p>The Proposition above is a consequence of Hamm's Lefschetz Theorem. For further details, see Beltrametti-Sommese, <em>The adjunction theory of complex projective varieties</em>, Corollary 2.3.4 page 51.</p> <p><strong>Remark.</strong> Given an integer $k \geq 0$, a line bundle $L$ on a projective variety $X$ is said to be $k$-ample if $a L$ is spanned for some $a > 0$, and the morphism $X \to \mathbb{P}^{h^0(aL)-1}$ defined by $|a L|$ has all fibers of dimension $\leq k$. For $k=0$, this is one of the basic characterization of ampleness.</p> http://mathoverflow.net/questions/120131/picard-group-of-a-very-ample-divisor-in-a-smooth-variety-of-dimension-3/120223#120223 Answer by Jason Starr for Picard group of a very ample divisor in a smooth variety of dimension >3 Jason Starr 2013-01-29T15:44:40Z 2013-01-29T15:44:40Z <p>Francesco Polizzi's answer is perfectly correct; also there are generalizations in the second part of <I>Stratified Morse Theory</I> by Goresky-MacPherson. However I want to point out that there is a different, much more algebraic perspective that works over an arbitrary field developed in <I>SGA 2. Cohomologie Locale des Faisceaux Coherents et Theoremes de Lefschetz Locaux et Globaux</I>, A. Grothendieck with an expose by Mme. M. Raynaud. The relevant result is Corollaire 3.6, Exp. XII, p. 153. </p>