Rationality is not a constructible property in families. In this article, we consider stronger notions of rationality and study their behavior in families of Fano varieties. We first show that being toric is a constructible property in families of Fano varieties. The second main result of this article concerns an intermediate notion that lies between toric and rational varieties, namely cluster type varieties. A cluster type & Qopf;-factorial Fano variety contains an open dense algebraic torus, but the variety does not need to be endowed with a torus action. We prove that, in families of & Qopf;-factorial terminal Fano varieties, being of cluster type is a constructible condition. As a consequence, we show that there are finitely many smooth families parametrizing n-dimensional smooth cluster type Fano varieties.

We state a correction of an error in the n = 4 n=4 case of Theorem 3 of the paper [L. Esser, L. Ji and J. Moraga, Symmetries of Fano varieties, J. Reine Angew. Math. 819 2025, 89-133].

We study rationality questions for Fano schemes of linear spaces on a smooth complete intersection X of two quadrics, especially over a non-closed field. Our approach is to study hyperbolic reductions of the pencil of quadrics associated to X. We prove that the Fano schemes Fr(X) of r-planes are birational to symmetric powers of hyperbolic reductions, generalizing results of Reid and Colliot-Th & eacute;l & egrave;ne-Sansuc-Swinnerton-Dyer, and we give several applications to rationality properties of Fr(X). For instance, we show that if X contains an (r+1)-plane over a field k, then Fr(X) is rational over k. When X has odd dimension, we show a partial converse for rationality of the Fano schemes of second maximal linear spaces, generalizing results of Hassett-Tschinkel and Benoist-Wittenberg. When X has even dimension, the analogous result does not hold, and we further investigate this situation over the real numbers. In particular, we prove a rationality criterion for the Fano schemes of second maximal linear spaces on these even-dimensional complete intersections over R; this may be viewed as extending work of Hassett-Koll & aacute;r-Tschinkel.

We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we give characterizations of rationality using the topology of the real locus and the intermediate Jacobian torsor obstruction of Hassett--Tschinkel and Benoist--Wittenberg. The double cover models we consider were introduced and previously studied by S. Frei, S. Sankar, B. Viray, I. Vogt, and the first author.

We undertake a study of conic bundle threefolds $\pi\colon X\to W$ over geometrically rational surfaces whose associated discriminant covers $\tilde{\Delta}\to\Delta\subset W$ are smooth and geometrically irreducible. First, we determine the structure of the group $\mathrm{CH}^2 X_{\overline{k}}$ of rational equivalence classes of curves. Precisely, we construct a Galois-equivariant group homomorphism from $\mathrm{CH}^2X_{\overline{k}}$ to a group scheme associated to the discriminant cover $\tilde{\Delta}\to \Delta$ of $X$. The target group scheme is a generalization of the Prym variety of $\tilde{\Delta}\to\Delta$ and so our result can be viewed as a generalization of Beauville's result that the algebraically trivial curve classes on $X_{\overline{k}}$ are parametrized by the Prym variety. We apply our structural result on curve classes to study the refined intermediate Jacobian torsor (IJT) obstruction to rationality introduced by Hassett--Tschinkel and Benoist--Wittenberg. The first case of interest is $W = \mathbb P^2$ and $\Delta$ is a smooth plane quartic. In this case, we show that the IJT obstruction characterizes rationality when the ground field has less arithmetic complexity (precisely, when the $2$-torsion in the Brauer group of the ground field is trivial). We also show that a hypothesis of this form is necessary by constructing, over any $k \subset\mathbb R$, a conic bundle threefold with $\Delta$ a smooth quartic where the IJT obstruction vanishes, yet $X$ is irrational over $k$.

We describe the 6-dimensional compact K-moduli space of Fano threefolds in deformation family No 2.18. These Fano threefolds are double covers of $\mathbb P^1\times\mathbb P^2$ branched along smooth $(2,2)$-surfaces, and Cheltsov--Fujita--Kishimoto--Park proved that any smooth Fano threefold in this family is K-stable. A member of family No 2.18 admits the structures of a conic bundle and a quadric surface bundle. We prove that K-polystable limits of these Fano threefolds admit conic bundle structures, but not necessarily del Pezzo fibration structures. We study this K-moduli space via the moduli space of log Fano pairs $(\mathbb P^1\times\mathbb P^2, c R)$ for $c=1/2$ and $R$ a $(2,2)$-divisor, which we construct using wall-crossings. In the case where the divisor is proportional to the anti-canonical divisor, the first author, together with Ascher and Liu, developed a framework for wall crossings in K-moduli and proved that there are only finitely many walls, which occur at rational values of the coefficient $c$. This paper constructs the first example of wall-crossing in K-moduli in the non-proportional setting, and we find a wall at an irrational value of $c$. In particular, we obtain explicit descriptions of the GIT and K-moduli spaces (for $c \leq 1/2$) of these $(2,2)$-divisors. Furthermore, using the conic bundle structure, we study the relationship with the GIT moduli space of plane quartic curves.

A double cover Y of ℙ^1 ×ℙ^2 ramified over a general (2,2)-divisor will have the structure of a geometrically standard conic bundle ramified over a smooth plane quartic Δ⊂ℙ^2 via the second projection. These threefolds are rational over algebraically closed fields; however, over nonclosed fields, including ℝ, their rationality is an open problem. In this paper, we characterize rationality over ℝ when Δ(ℝ) has at least two connected components (extending work of M. Ji and the second author) and over local fields when all odd degree fibers of the first projection have nonsquare discriminant. We obtain these applications by proving general results comparing the conic bundle structure on Y with the conic bundle structure on a well-chosen intersection of two quadrics. The difference between these two conic bundles is encoded by a constant Brauer class, and we prove that this class encodes the obstruction to the existence of a section of the first projection Y→ℙ^1.

We show that if X ⊂ P k N X\subset \mathbb P^N_k is a normal variety of dimension ≥ 3 \geq 3 and H ⊂ P k N H\subset \mathbb P^N_k a very general hypersurface of degree d = 4 d=4 or ≥ 6 \geq 6 , then the restriction map Cl ⁡ ( X ) → Cl ⁡ ( X ∩ H ) \operatorname {Cl}(X)\to \operatorname {Cl}(X\cap H) is an isomorphism up to torsion. If dim ⁡ X ≥ 4 \dim X\geq 4 , the result holds for d ≥ 2 d\geq 2 . The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.

In this note we construct an example of a smooth projective threefold that is irrational over ℚ but is rational at all places. Our example is a complete intersection of two quadrics in ℙ^5 , and we show it has the desired rationality behavior by constructing an explicit element of order 4 in the Tate–Shafarevich group of the Jacobian of an associated genus 2 curve.

We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k \leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand, we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic upper bound is not expected to be sharp. We obtain sharp bounds for certain classes of varieties. For toric varieties, we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete intersections, we prove the asymptotic equality $\lim_{n\to \infty} m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the maximally symmetric ones and show that they are closely related to the Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut out by Fermat hypersurfaces. Finally, we draw a connection between maximally symmetric Fano varieties and boundedness of Fano varieties. For instance, we show that the class of $S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the $S_7$-equivariant Fano $4$-folds are unbounded.

Koll\'ar proved that a very general $n$-dimensional complex hypersurface of degree at least $3\lceil (n+3)/4\rceil$ is not birational to a fibration in rational curves. This is most interesting when the hypersurface is Fano, in which case it is covered by rational curves. In this paper, we extend Koll\'ar's ideas and show that for any genus $g$, there are Fano hypersurfaces (in more restrictive degree and dimension ranges) that are not birational to fibrations in genus $g$ curves. In other words, we show that the fibering genus of these hypersurfaces can be arbitrarily large. The fibering genus of a variety has been studied in work of Konno, Ein--Lazarsfeld, and Voisin, but this is the first paper to explore these ideas in the Fano range. Following Koll\'ar, we degenerate to characteristic $p>0$ to rule out these fibrations. A crucial input is Tate's genus change formula and its generalizations, which imply that any regular curve of genus $g$ is smooth if $p$ is sufficiently large compared to $g$.

We explain a proof of the Theorem of the Base: the Neron– Severi group of a proper variety is a finitely generated abelian group. We discuss, quite generally, the Picard functor and its torsion and identity components. We study representability and finiteness properties of the Picard functor, both absolutely and in families. Along the way, we streamline the original proof by using alterations, and we discuss some examples of peculiar Picard schemes.

We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue characteristic cannot specialize to the identity. As an application, we show that very general $n$-dimensional hypersurfaces of degree $d \geq 5 \lceil (n+3)/6 \rceil$ have no finite order birational automorphisms.

We prove a structural result for geometrically non-reduced varieties and give applications to Fano varieties. For example, we show that if X is the generic fibre of a Mori fibre space of relative dimension n, and the characteristic is p > 2n + 1, then any geometric non-reducedness of X comes from the base of some fibration.

Let $T$ be a complete equicharacteristic local (Noetherian) UFD of dimension $3$ or greater. Assuming that $|T| = |T/m|$, where $m$ is the maximal ideal of $T$, we construct a local UFD $A$ whose completion is $T$ and whose formal fibers at height one prime ideals have prescribed dimension between zero and the dimension of the generic formal fiber. If, in addition, $T$ is regular and has characteristic zero, we can construct $A$ to be excellent.

Let T be a complete local (Noetherian) equidimensional ring with maximal ideal m such that the Krull dimension of T is at least two and the depth of T is at least two. Suppose that no integer of T is a zerodivisor and that |T|=|T/m|. Let d and t be integers such that 1 $\leq$ d $\leq$ dimT-1, 0 $\leq$ t $\leq$ dimT - 1, and d - 1 $\leq$ t. Assume that, for every p in AssT, ht(p) $\leq$ d-1 and that if z is a regular element of T and Q is in Ass(T/zT), then ht(Q) $\leq$ d. We construct a local unique factorization domain A such that the completion of A is T and such that the dimension of the formal fiber ring at every height one prime ideal of A is d - 1 and the dimension of the formal fiber ring of A at (0) is t.