Search

The goal of this dissertation is twofold. First, we give a proof of the geometric cone conjecture in relative dimension at most $2$ in the full generality of klt CY pairs. This generalizes previous work done by Kawamata in \cite{Kaw97}, Totaro in \cite{Tot10}, and Filipazzi, Hacon, and Svaldi in \cite{FHS21}. A corollary to this result is that all klt pairs $(X.\Delta)$ with $\kappa(X,\Delta)\geq \dim X - 2$ have only finitely many isomorphism classes of minimal models. In the second part, we prove that the property of Bott vanishing fails for 68 deformation families of smooth Fano threefolds in characteristic $0$. This result is sharp, as subsequent work by Totaro in \cite{Tot24} shows that Bott vanishing is satisfied by the remaining 37 deformation families.

The Cox ring of an algebraic variety encodes important information on the birational geometry of the variety. When its Cox ring is finitely generated, a variety admits particularly desirable properties in the context of the Minimal Model Program and is called a Mori dream space. For example, all toric varieties are known to be Mori dream spaces so a natural next step in the problem is to study the birational geometry of projective varieties that can be constructed as blowups of toric varieties by studying their pseudoeffective cones and Cox rings. In this dissertation, we present a concrete criterion for the finite generation of the Cox ring of toric projective surfaces of Picard number one blown up at a smooth point using the coordinates of a polytope of the toric variety. We also present a criterion for the irreducibility of an effective divisor of the moduli space of n-pointed stable rational curves.

In this paper we prove the birational superrigidity of Fano-Mori fibre spaces $\pi\colon V\to S$, every fibre of which is a complete intersection of type $d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$, satisfying certain conditions of general position, under the assumption that the fibration $V/S$ is sufficiently twisted over the base (in particular, under the assumption that the $K$-condition holds). The condition of general position for every fibre guarantees that the global log canonical threshold is equal to one. This condition bounds the dimension of the base $S$ by a constant that depends on the dimension $M$ of the fibre only (as the dimension $M$ of the fibre grows, this constant grows as $\frac12 M^2$). The fibres and the variety $V$ itself may have quadratic and bi-quadratic singularities, the rank of which is bounded from below., Comment: 86 pages, the final version

In this note, we study base point freeness up to taking p-power, which we will call p-power freeness. We first establish some criteria for p-power freeness as analogues of criteria for semi-ampleness. We then apply these results to three-dimensional birational geometry.

We rework the Mori-Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians., Comment: Minor changes. To appear in Rev. Mat. Complut

One of the main research programs in Algebraic Geometry is the classification of varieties. Towards this goal two methodologies arose, the first is classifying varieties up to isomorphism which leads to the study of moduli spaces and the second is classifying varieties up to birational equivalences which leads to the study of birational geometry. Part of the engine of the birational classification is the Minimal Model Program which, given a variety, seeks to find "nice" birational models, which we call minimal models. Towards this direction much progress has been made but there is also much to be done. One aspect of interests is the role of algebraic fiber spaces as the end results of the Minimal Model Program are categorized into Mori fiber spaces, Iitaka fibrations over canonical models and varieties of general type. A natural problem to consider is, starting with an algebraic fiber space, how might it behave with respect to the Minimal Model Program. For case of elliptic threefolds, it was shown by Grassi, that minimal models of elliptic threefolds relate to log minimal models of the base surface. This shows that minimal models, in a sense, have to respect the fiber structure for elliptic threefolds. In this dissertation, I will provide a framework towards a generalization for higher dimensional elliptic fibration and along the way recover the results of Grassi for elliptic fourfolds with section.

Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0, Comment: 9 pages

We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_{(n)}^{(l)}$, which is the quotient of the product of a certain curve $C_{(n)}$ by itself $l$ times by a group $G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right )^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_{(n)}^{(l)}$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$, its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model that admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

Consider the blow-up $X$ of $\mathbb{P}^3$ at 6 points in very general position and the 15 lines through the 6 points. We construct an infinite-order pseudo-automorphism $\phi_X$ on $X$, induced by the complete linear system of a divisor of degree 13. The effective cone of $X$ has infinitely many extremal rays and hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $\phi_X$ on $S$ realizes one of Keum's 192 infinite-order automorphisms of Jacobian K3 Kummer surfaces. In general, we show the blow-up of $\mathbb{P}^n$ ($n\geq 3$) at $(n+3)$ very general points and certain 9 lines through them is not Mori Dream, with infinitely many extremal effective divisors. As an application, for $n\geq 7$, the blow-up of $\overline{M}_{0,n}$ at a very general point has infinitely many extremal effective divisors., Comment: We added the omitted part of the proof of Prop. 12.1. To appear in IMRN

We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X-->Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a non-zero, non-big movable divisor on X. Our main result is that if Y is not P^1 or P^2, then the Picard number rho(X) of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X-->Z, regular and proper on an open subset of X, with dim(Z)=3, then either X is a product of surfaces, or rho(X) is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry., 25 pages. Minor changes. To appear in Algebra & Number Theory

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n$. We investigate positivity properties of the spaces $R_e(X)$ of degree $e$ rational curves in $X$. We show that for small $e$, $R_e(X)$ has no rational curves meeting the locus of smooth embedded curves. We show that for $n \leq d$, there are no rational curves in the locus $Y \subset X$ swept out by lines. And we exhibit differential forms on a smooth compactification of $R_e(X)$ for every $e$ and $n-2 \geq d \geq \frac{n+1}{2}$., Comment: revised version

We develop some foundational results in a higher dimensional foliated Mori theory, andshow how these results can be used to prove a structure theorem for the Kleiman-Mori coneof curves in terms of the numerical properties of $K_{\cal F}$ for rank 2 foliationson threefolds. We also make progresstoward realizing a minimal model program for rank 2 foliations on threefolds.

This volume carries 10 papers related to the intra-disciplinary workshopGeometry at Large,which took place on Fuerteventura island (Spain) in December 2018. This workshop, supported by the EPSRC Programme Grant Symmetries and Correspondences, involved leading researchers working in adelic, affine, algebraic, anabelian, arithmetic, birational, complex, derived, enumerative, local, non-commutative, motivic, model theoretical, padic, real, symplectic, and tropical geometries

By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-$2$ $K3$ surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-$4$ $K3$ surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to $D$-lattices. In particular, for the $19$-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-$4$ $K3$ surfaces. The analogous $18$-dimensional case, which corresponds to hyperelliptic degree-$4$ $K3$ surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.

We solve the infinitesimal Torelli problem for $3$-dimensional quasi-smooth ${\mathbb{Q}}$-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively distribute themselves in examples and counterexamples for the infinitesimal Torelli claim and which share the analogue, and in some cases the same, Hodge-diagram properties as the length $3$ Gushel-Mukai chain of prime smooth Fanos of coindex $3$ and degree $10$., Comment: Final version, to appear in Journal of Geometry and Physics

Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C} \mathbb{P}^n$ of degree $d$. We prove the de Jong-Debarre Conjecture for $n \geq 2d-4$: the space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result for $k$-planes: if $n \geq 2 \binom{d+k-1}{k} + k$, then the space of $k$-planes on $X$ will be irreducible of the expected dimension. As applications, we prove that an arbitrary smooth hypersurface satisfying $n \geq 2^{d!}$ is unirational, and we prove that the space of degree $e$ curves on $X$ will be irreducible of the expected dimension provided that $d \leq \frac{e+n}{e+1}$., Comment: Updated version

In this paper we study the pluricanonical maps of minimal projective 3-folds of general type with geometric genus \begin{document}$ 1 $\end{document} , \begin{document}$ 2 $\end{document} and \begin{document}$ 3 $\end{document} . We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of \begin{document}$ \varphi_{16} $\end{document} , \begin{document}$ \varphi_{6} $\end{document} and \begin{document}$ \varphi_{5} $\end{document} respectively.

Let $X$ be a Hirzebruch surface, and let $H$ be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves $M_{X,H}(r,c_1,c_2)$ is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one. We then study sharp Bogomolov inequalities $\Delta \geq \delta_H(c_1/r)$ for the discriminants of stable sheaves which take the polarization and slope into account; these inequalities essentially completely describe the characters of stable sheaves. The function $\delta_H(c_1/r)$ can be computed to arbitrary precision by a limiting procedure. In the case of an anticanonically polarized del Pezzo surface, exceptional bundles are always stable and $\delta_H(c_1/r)$ is computed by exceptional bundles. More generally, we show that for an arbitrary polarization there are further necessary conditions for the existence of stable sheaves beyond those provided by stable exceptional bundles. We compute $\delta_H(c_1/r)$ exactly in some of these cases. Finally, solutions to the existence problem have immediate applications to the birational geometry of moduli spaces of sheaves., Comment: 70 pages, 9 figures, comments welcome! v2: minor corrections

In this paper we study the geometry of GIT configurations of $n$ ordered points on $\mathbb{P}^1$ both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient $(\mathbb{P}^1)^n//PGL(2)$, taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of $(\mathbb{P}^1)^n//PGL(2)$ in its natural embedding, and its group of automorphisms., Comment: 22 pages (fixed one small inaccuracy)

Extending work of Meinhardt and Partsch, we prove that two varieties are isomorphic away from a subset of a given dimension if and only if certain quotients of their categories of coherent sheaves are equivalent. This result interpolates between Gabriel’s reconstruction theorem and the fact that two varieties are birational if and only if they have the same function field.

We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions., 40 pages

The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein $3$-fold quotient singularities $\mathbb{A}^3/G$ with the representation theory of the group $G$. The first crepant resolution studied in depth was the $G$-Hilbert scheme $G\text{-Hilb}\,\mathbb{A}^3$, which is also a moduli space of $\theta$-stable representations of the McKay quiver associated to $G$. As the stability parameter $\theta$ varies, we obtain many other crepant resolutions. In this paper we focus on the case where $G$ is abelian, and compute explicit inequalities for the chamber of the stability space defining $G\text{-Hilb}\,\mathbb{A}^3$ in terms of a marking of exceptional subvarieties of $G\text{-Hilb}\,\mathbb{A}^3$ called Reid's recipe. We further show which of these inequalities define walls. This procedure depends only on the combinatorics of the exceptional fibre and has applications to the birational geometry of other crepant resolutions.

This is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry. In doing so, the main theme of the article is the following: there are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. These scenarios are somewhat unexpected and have had a profound impact in mathematics., 10 pages, 3 figues. To appear in the Notices of the American Mathematical Society

We introduce equivariant Burnside groups, new invariants in equivariant birational geometry, generalizing birational symbols groups for actions of finite abelian groups, due to Kontsevich, Pestun, and the second author, and study their properties. We establish a specialization map for the equivariant birational type of a smooth algebraic variety with an action of a finite group., 41 pages

Let $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $\alpha$-semistable pairs on $\mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincar\'e polynomial of the compactified space., Comment: Comments are welcome

In this note, we survey some recent developments in birational geometry concerning the boundedness of algebraic varieties. We delineate a strategy to extend some of these results to the case of generalized pairs, first introduced by Birkar and Zhang, when the associated log canonical divisor is ample, and the volume is fixed. In this context, we show a version of deformation invariance of plurigenera for generalized pairs. We conclude by discussing an application to the boundedness of varieties of Kodaira dimension $\kappa(X)=\dim(X)-1$., Comment: Final accepted version. Comments are welcome

Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kahler base B. Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let La be convex line bundles over B, Aa smooth divisors of B arising as the zero loci of generic sections of La , and a particular fixed-point section of E. Further assume the {Aa} to be mutually disjoint. The manifold is a new manifold with tautological line bundles over new projective spaces in the geometry, where previously there was a simpler vector bundle in the given local geometry (Section 1.5). Thus, we compute genus-0 Gromov-Witten invariants of in terms of genus-0 Gromov-Witten invariants of B and of {Aa}, the matrix used for the symplectic reduction description of the fiber of the toric fibration E→B, and the restriction maps . The proofs utilize the fixed-point localization technique describing the geometry of and its genus-0 Gromov-Witten theory, as well as the Quantum Lefschetz theorem relating the genus-0 Gromov-Witten theory of A with that of B.

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace. Part of the proof uses the fact that subvarieties of positive codimension are stably displaceable., Comment: 111 pages, 23 figures. Improved exposition and added a few more figures

In this paper, we prove that the moduli space of stable pairs (or sheaves) on del Pezzo surface S is isomorphic to the Maruyama type transformation of a projective bundle of a quiver representation...

For a Zariski general (regular) hypersurface $V$ of degree $M$ in the $(M+1)$-dimensional projective space, where $M$ is at least 16, with at most quadratic singularities of rank at least 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that $V$ is non-rational and its groups of birational and biregular automorphisms coincide. The set of non-regular hypersurfaces has codimension at least $\frac12(M-11)(M-10)-10$ in the natural parameter space., Comment: 39 pages

Building on the work of Mukai, we explore the birational geometry of the moduli spaces $$M_{S,L}$$ of semistable rank two torsion-free sheaves, with $$c_1=-K_S$$ and $$c_2=2$$ , on a polarized degree one del Pezzo surface (S, L); this is related to the birational geometry of the blow-up X of $$\mathbb {P}^4$$ in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space $$Y=M_{S,-K_S}$$ , which is a remarkable family of smooth Fano fourfolds. In particular we describe the relevant cones of divisors of Y, the group of automorphisms, and the base loci of the anticanonical and bianticanonical linear systems.

We use a recent advance in birational geometry to prove new lower bounds on the essential dimension of some finite groups.

In this paper, we consider the class of first-order algebraic ordinary differential equations (AODEs), and study their rational general solutions. A rational general solution contains an arbitrary constant. We give a decision algorithm for finding a rational general solution, in which the arbitrary constant appears rationally, of the whole class of first-order AODEs. As a byproduct, this leads to an algorithm for determining a rational general solution of a class of first-order AODE which covers almost all first-order AODEs from Kamke's collection. The method is based intrinsically on the consideration of the AODE from a geometric point of view. In particular, parametrizations of algebraic curves play an important role for a transformation of a parametrizable first-order AODE to a quasi-linear differential equation.

In the present paper and the companion paper (Berman, 2017) a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and shown to converge in probability towards a canonical measure, coinciding with the canonical measure of Song-Tian and Tsuji. In the case of a variety X of general type we obtain as a corollary that the (possibly singular) Kahler-Einstein metric on X with negative Ricci curvature is the limit of a canonical sequence of quasi-explicit Bergman type metrics. In the opposite setting of a Fano variety X we relate the canonical point processes to a new notion of stability, that we call Gibbs stability, which admits a natural algebro-geometric formulation and which we conjecture is equivalent to the existence of a Kahler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.

Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces. We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces. The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algebraic Geometry in Utah, July 2015., Comment: 25 pages, 3.5 figures. v2: expanded comparison to Lazarsfeld's methods and results; addressed referee comments

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general ν-gonal curve. We classify its irreducible components having at least expected dimension. We moreover describe the general member F of such components just in terms of extensions of line bundles with suitable "minimality properties", providing information on the birational geometry of such components as well as on the very-ampleness of F.

Assuming Lang's conjecture, we prove that for a fixed prime $p$, number field $K$, and positive integer $g$, there is an integer $r$ such that no principally polarized abelian variety $A/K$ of dimension $g$ has full level $p^r$ structure. To this end, we use a result of Zuo to prove that for each closed subvariety $X$ in the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$, there exists a level $m_X$ such that the irreducible components of the preimage of $X$ in $\mathcal{A}_g^{[m]}$ are of general type for $m > m_X$., 17 pages. References to new work of Brunebarbe added; discussion of implications arising from Lang's geometric conjecture suppressed in light of Brunebarbe's new results. Section 4 recast in more general terms; see Proposition 4.3 and Theorem 1.13

We prove that the moduli spaces of rational curves of degree at most 3 in linear sections of the Grassmannian G r ( 2 , 5 ) are all rational varieties. We also study their compactifications and birational geometry.

Hassett constructed a class of modular compactifications of Mg,n by adding weights to the marked points. This leads to a natural wall and chamber decomposition of the domain of admissible weights D...

Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.

Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a “web of flops” provided by the Grothendieck resolution associated to a reductive Lie algebra $$\mathfrak {g}$$ and study the corresponding schober-type diagram. For $$\mathfrak {g}={\mathfrak {s}\mathfrak {l}}_3$$ we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.

We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle., Comment: 14 pages

We consider the space $\mathcal R_{g,S_3}^{S_3}$ of curves with a connected $S_3$-cover, proving that for any odd genus $g\geq 13$ this moduli is of general type. Furthermore we develop a set of tools that are essential in approaching the case of $G$-covers for any finite group $G$., 32 pages, 1 figure

We continue previous works by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions, by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brinzanescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational., Comment: to appear in Tran. A.M.S

A Cayley octad is a set of 8 points in P3 which are the base locus of a net of quadrics. Blowing up the points of the octad gives a morphism to P2 defined by the net; the fibres of this morphism are intersections of two quadrics in the net, hence curves of genus 1. The generic fibre therefore has a group structure, and the action of this group on itself extends to a birational action on the whole variety. In particular, if the generic fibre has a large group of rational points, the birational automorphism group, and hence the birational geometry, of the variety must be complicated. It is natural to ask whether the converse is true: if the generic fibre has only a small group of rational points, is the birational geometry of the variety correspondingly simple? In this paper we study a special Cayley octad with the property that the generic fibre has finitely many rational points. In Section 1 we find that such an octad only exists in characteristic 2, and is unique up to projective transformations. Our main results then show that the simplicity of the generic fibre is indeed reected in the simplicity of the birational geometry of our blowup. In Section 2 we show that the cones of nef and movable divisors are rational polyhedral, as predicted by the Morrison{Kawamata conjecture. Finally, in Section 3 we prove that our blowup has the \best possible" birational geometric properties: it is a Mori dream space.