In a paper of Bloch (Torsion algebraic cycles and a theorem of Roitman) he claims the following:


Let $X$ be a projective smooth variety over an algebraic closed field. Now, let $Y$ be a general linear space section of large degree and dimension $2$. Then $\text{Alb}(X)\cong \text{Alb}(Y)$.


Why is this true? Is there any reference for this?


Thank you.



An isomorphism between the Albanese varieties of a variety and a general linear space section
I am expanding on my comments above. I looked at Bloch's paper. He includes a further hypothesis on $Y$, namely that it contains a reducible curve whose irreducible components are smooth and whose union has only ordinary double points (nodes) as singularities. That does not really affect the claim.


Let $X$ be a projective, Cohen-Macaulay scheme of pure dimension $r\geq 3$ over a field $k$. Let $\mathcal{O}(1)$ be an ample invertible sheaf on $X$. Assume that the the $k$-smooth locus $X\_{\text{sm}}$ is a dense open whose complement $X\_{\text{sing}}$ has codimension $\geq 4$ in $X$ at every point. Since $X$ is Cohen-Macaulay, there is a coherent dualizing sheaf $\omega\_{X/k}$. By Serre vanishing, there exists an integer $n\_0\geq 1$ such that for every $n\geq n\_0$ and for every $q\geq 1$, the cohomology group $$H^q(X,\omega\_X\otimes\_{\mathcal{O}\_X}\mathcal{O}(n))$$ is the zero group. Let $Y\subset X$ be the common zero scheme of a system $(t\_1,\dots,t\_c)$ of sections $t\_i\in H^0(X,\mathcal{O}(n\_i))$, with $c\leq \text{dim}(X)-2$. The following result is a special case of several results from SGA 2.


MR2171939 (2006f:14004)   
 
Grothendieck, Alexander   

Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2).   
 
Séminaire de Géométrie Algébrique du Bois Marie, 1962. Augmenté d'un exposé de Michèle Raynaud.   

With a preface and edited by Yves Laszlo. Revised reprint of the 1968 French original.   

Documents Mathématiques (Paris), 4. Société Mathématique de France, Paris, 2005. x+208 pp. ISBN: 2-85629-169-4   



**Theorem. [SGA 2, approx. Corollaire XII.3.6.]** If $(t\_1,\dots,t\_c)$ is a regular system, if every $n\_i \geq n\_0$, if $Y$ contains every singular point of $X$ that is worse than a local complete intersection singularity, and if $\text{dim}(Y)\geq 2$, resp. $\text{dim}(Y)\geq 3$, then the restriction map of Picard schemes, $\text{Pic}\_{X/k}\to \text{Pic}\_{Y/k}$, is an open immersion, resp. it is an isomorphism. In particular, the induced map of identity components is an isomorphism. The dual map of identity components is an isomorphism of Albanese schemes.


**Proof.** Under the hypotheses, by Corollaire XII.3.6 of SGA 2, the restriction map is injective on geometric points, resp. it is an isomorphism on geometric points. Thus, to prove that the restriction map is an open immersion, resp. an isomorphism, it suffices to prove that the restriction map is formally étale. By induction on $c$, it suffices to consider the case when $c$ equals $1$, i.e., when $Y$ is a hypersurface in $X$.


Given an infinitesimal lifting problem for an invertible sheaf on $X$, assuming the existence of a lifting on $Y$, the obstruction to finding a compatible lifting on $X$ is an element of $$H^2(X,\mathcal{O}\_X(-\underline{Y})) = H^2(X,\mathcal{O}(-n\_1)).$$ By Serre Duality, this group is the same as $H^{r-2}(X,\omega\_{X/k}(n\_1))$. By the hypothesis that $r\geq 3$ and the hypothesis that $n\_1\geq n\_0$, this obstruction group is the zero group. Thus, there exists an infinitesimal lifting. It follows that the restriction map is formally smooth.


To prove that the restriction map is also formally unramified, hence formally étale, it suffices to prove vanishing of the relative differentials. That comes to vanishing of $H^1(X,\mathcal{O}\_X(-\underline{Y}))$, i.e., vanishing of the Serre Dual cohomology group $H^{r-1}(X,\omega\_{X/k}(n\_1))$, which follows by the same argument. **QED**


**Nota bene. (1).** Surjectivity of the induced restriction map on component groups often fails if $\text{dim}(Y)$ equals $2,$ e.g., for smooth surfaces in $\mathbb{P}^3$ of degree $d$ that happen to contain a line. The setup in Bloch's argument, where he chooses $Y$ to contain a specified reducible curve in $X$, is very likely to give a surface where surjectivity on component groups fails. Results that prove surjectivity of the restriction map on component groups if $\text{dim}(Y)$ equals $2$ are often called "Noether-Lefschetz Theorems." One such results is Nori's Connectedness Theorem. Since the Albanese scheme is defined in terms of the dual of the **identity component** of the Picard scheme rather than the full Picard group, this failure of surjectivity for the component groups is irrelevant for the application to Albanese varieties. 


**(2).** If $Y$ fails to contain every non-LCI singularity of $X$ (or rather, every non-parafactorial singularity), then the conclusion can easily fail. For instance, let $Y\subset \mathbb{P}^m$ be an Abelian variety of dimension $r-1\geq 2$ embedded by the complete linear system of a very ample invertible sheaf. Let $X\subset \mathbb{P}^{m+1}$ be the projective cone over $Y$. Then $\text{Pic}(X)$ equals $\mathbb{Z}$, so the restriction is not formally smooth. The scheme $Y$ fails to contain the vertex of the cone, and this is a point where $X$ fails to be (locally) parafactorial. 


