William Edge was born in Stockport on 8th November 1904 to parents who were both schoolteachers. After a secondary education at the local Stockport Grammar School, in 1923 he went up to Cambridge to study mathematics at Trinity College. After graduation, he became a Ph.D. student of Henry Baker, where his dissertation generalised Luigi Cremona’s results about ruled surfaces in the real projective space. His appointment as a research fellow in Trinity College ensued in 1928. After 4 years he secured a lectureship (assistant professorship) at the University of Edinburgh, where he remained for the rest of his career. Elected a Fellow of the Royal Society of Edinburgh 2 years later in 1932, Edge was later promoted to reader (associate professor) in 1949. However, it took until 1969 for him to be appointed to a full professorship, 6 years prior his retirement in 1975. Edge never married, he had no children, he never drove a car, he was reluctant to travel, and he disdained radio and television. Apart from mathematics, Edge loved hill walking, singing and playing the piano. He spent his final years in the retirement house in Bonnyrigg near Edinburgh, where he died on 27 September 1997.
William Edge’s scientific results influenced the research of many mathematicians worldwide. His papers, even from 50 years ago, continue to attract attention. This volume is dedicated to the memory of William Edge.
To carry Edinburgh’s tradition in geometry into the future, Ahmadinezhad, Cheltsov, Guletskiĭ, Kaloghiros, Logvinenko and Testa organised five Edge Days workshops at Edinburgh in 2013–2017. Among their participants were most of contributors to this volume as well as Caucher Birkar, Alexey Bondal, Alessio Corti, Ruadhai Dervan, Yoshinori Gongyo, Liana Heuberger, Ilia Itenberg, Dmitry Kaledin, Masayuki Kawakita, Alvaro Liendo, Angelo Lopez, Daniel Loughran, Frédéric Mangolte, Takuzo Okada, John Ottem, Dmitri Panov, Elisa Postinghel, Francesco Russo, Edoardo Sernesi, Nicholas ShepherdBarron, Alexei Skorobogatov and Yuri Tschinkel.
The second part of the Edge Volume contains 23 research papers, whose authors are Florin Ambro, Asher Auel, Artem Avilov, Edoardo Ballico, Christian Böhning, HansChristian Graf von Bothmer, Harry Braden, Aiden Bruen, Ivan Cheltsov, Giulio Codogni, Igor Dolgachev, David Eklund, Philippe Ellia, Andrea Fanelli, Benson Farb, Ivan Fesenko, Claudio Fontanari, Juan FríasMedina Patricio Gallardo, Marat Gizatullin, Vladimir Guletskiĭ, David Holmes, Martin Kalck, Joseph Karmazyn, Jesse Leo Kass, Ludmil Katzarkov, Eduard Looijenga, Lisa Marquand, Jesus MartinezGarcia, James McQuillan, Nicola Pagani, Jihun Park, Alena Pirutka, Yuri Prokhorov, Costya Shramov, Leonardo Soriani, Roberto Svaldi, Luca Tasin, Alessandro Verra, Sergei Vostokov, Jonas Wolter, Joonyeong Won, Seok Ho Yoon, Misha Zaidenberg, Alexis Zamora, Zheng Zhang, and a postface by James Hirschfeld. Most of them were participants of Edge Days, while the others are mathematicians who personally knew William Edge or used his work in their research.
Let us briefly describe the scientific content of the papers in the second part of the volume.
In the paper On toric face rings I, Florin Ambro constructs a Deligne–Du Bois complex for algebraic varieties which are locally isomorphic to the spectrum of a toric face ring.
In the paper Stable rationality of quadric and cubic surface bundle fourfolds, Asher Auel, Christian Böhning and Alena Pirutka study the stable rationality problem for quadric and cubic surface bundles over surfaces. Their main result is the following
Theorem
A very general hypersurface of bidegree (2, 3) in Open image in new window is not stably rational.
This result provides another example of a smooth family of rationally connected fourfolds with rational and nonrational fibers. The paper also contains examples of a quadric surface bundle over Open image in new window with discriminant curve of even degree (greater than 7) that have nontrivial unramified Brauer groups and universally \(\mathrm{CH}_0\)trivial resolutions.
In the paper Automorphisms of singular threedimensional cubic hypersurfaces, Artem Avilov studies singular threedimensional cubic hypersurfaces in Open image in new window faithfully acted on by a finite group. He proves the following
Theorem

the threefold X is the Segre cubic, Open image in new window , and G is one of the following subgroups: \(\mathfrak {A}_{5}\) (standard subgroup), \(\mathfrak {S}_{5}\) (standard subgroup), \(\mathfrak {A}_{6}\), \(\mathfrak {S}_{6}\);
 up to projective equivalence, the threefold X is given byOpen image in new window , and G is one of the following subgroups: Open image in new window , \(\mathfrak {S}_{3}^2\), Open image in new window ;$$\begin{aligned} x_{0}x_{1}x_{2}+x_{3}x_{4}(x_0+x_1+x_2+x_3+x_4)=0, \end{aligned}$$
 up to projective equivalence, the threefold X is given byand Open image in new window ;$$\begin{aligned} x_{0}x_{1}x_{2}+x_{1}x_{2}x_{3}&+x_{2}x_{3}x_{4}+x_{3}x_{4}x_{0}+x_{4}x_{0}x_{1} \\&=x_{0}x_{1}x_{3}+x_{1}x_{2}x_{4}+x_{2}x_{3}x_{0}+x_{3}x_{4}x_{1}+x_{4}x_{0}x_{2}, \end{aligned}$$
 up to projective equivalence, the threefold X is given byOpen image in new window , and either \(G\cong \mathfrak {S}_{3}^2\) or Open image in new window ;$$\begin{aligned} x_{0}x_{1}x_{2}x_{0}x_{1}x_{3}&+x_{0}x_{1}x_{4}+x_{0}x_{2}x_{3} 3x_{0}x_{2}x_{4}\\&+x_{0}x_{3}x_{4}x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4}=0, \end{aligned}$$
 up to projective equivalence, the threefold X is given by\(\mathrm{Aut}(X)\cong \mathfrak {S}_{5}\), and G is one of the following subgroups: Open image in new window , \(\mathfrak {A}_{5}\), \(\mathfrak {S}_{5}\);$$\begin{aligned} \sum \limits _{0\leqslant i<j<k\leqslant 4}\!\!\!\!x_{i}x_{j}x_{k}=0, \end{aligned}$$
 up to projective equivalence, the threefold X is given bywhere \(\omega \) is a primitive cubic root of unity, and Open image in new window .$$\begin{aligned} x_{0}x_{1}x_{2}+x_{1}x_{2}x_{3}&+x_{2}x_{3}x_{4}+x_{3}x_{4}x_{0}+x_{4}x_{0}x_{1} \\&=\omega \bigl (x_{0}x_{1}x_{3}+x_{1}x_{2}x_{4}+x_{2}x_{3}x_{0}+x_{3}x_{4}x_{1}+x_{4}x_{0}x_{2}\bigr ), \end{aligned}$$
A smooth connected curve C in Open image in new window is said to be of maximal rank if the natural restriction maps Open image in new window are either injective or surjective for every m. In the paper Maximal rank of space curves in the range A, Edoardo Ballico, Claudio Fontanari and Philippe Ellia prove the following
Theorem
There exists a constant K such that for all natural numbers d and g with \(g\leqslant Kd^{{3}/{2}}\) there exists an irreducible component of the Hilbert scheme of Open image in new window whose general element is a smooth connected curve of degree d and genus g of maximal rank.
Question
Is very general Gushel–Mukai fourfold irrational?
To study this question, Böhning and Bothmer study degenerations of Gushel–Mukai fourfolds that satisfy certain (strong) conditions, which are natural if one wants to apply degeneration technique of Voisin, Colliot–Thélène, Pirutka and Totaro to Gushel–Mukai fourfolds. However, they prove that such degenerations do not exist.
In the paper A canonical form for a symplectic involution, Harry Braden presents a canonical form for an involution in Open image in new window and applies his construction to Riemann surfaces.
In a projective plane over a field \(\mathbb {F}\), the diagonal points of a quadrangle are collinear if and only if \(\mathbb {F}\) has characteristic 2. Such a quadrangle together with its diagonal points and the lines connecting these points form the subplane of order 2, called a Fano plane. In the paper Desargues configurations with four selfconjugate points, Aiden Bruen and James McQuillan provide a similar type of synthetic criterion and construction for characteristic 3 fields. This fits perfectly the research interests of Edge, since he moved towards finite geometry after studying classical geometry.
Theorem
Here, the numbers \(\alpha (E_X,\mathrm{Diff}_{E_X}(0))\) and \(\alpha (E_Y,\mathrm{Diff}_{E_Y}(0))\) are \(\alpha \)invariants of Tian of the log Fano varieties \((E_X,\mathrm{Diff}_{E_X}(0))\) and \((E_Y,\mathrm{Diff}_{E_Y}(0))\), respectively.
In the paper A note on the fibres of Mori fibre spaces, Giulio Codogni, Andrea Fanelli, Roberto Svaldi and Luca Tasin study Fano varieties that can be realised as fibres of a Mori fibre space.
In the paper Generalised Kawada–Satake method for Mackey functors in class field theory, Ivan Fesenko, Sergei Vostokov and Seok Ho Yoon generalise Kawada–Satake method for Mackey functors in the class field theory of positive characteristic.
In the paper Some remarks on Humbert–Edge’s curves, Juan FríasMedina and Alexis Zamora explain Edge’s approach to the study of Humbert’s curves. They reprove some of his results and consider their generalisations.
In the paper Compactifications of the moduli space of plane quartics and two lines, Patricio Gallardo, Jesus MartinezGarcia and Zheng Zhang study the moduli space of triples \((C,L_1,L_2)\) that consists of a quartic curve C and two lines \(L_1\) and \(L_2\) in Open image in new window . They construct and compactify this moduli space in two ways: via geometric invariant theory and by using the period map of certain lattice polarized K3 surfaces.
In the paper Two examples of affine homogeneous varieties, Marat Gizatullin studies two explicit flexible affine homogeneous varieties that have infinitedimensional groups of automorphisms. In both cases, Gizatullin proves the existence of automorphisms that do not belong to the connected components of identity.
In the paper Chow motives of abelian type over a base, Vladimir Guletskiĭ generalises the theorem of Kimura about motives of smooth projective curves to the relative setting.
In the paper Extending the double ramification cycle using Jacobians, David Holmes, Jesse Leo Kass and Nicola Pagani prove that the extension of the double ramification cycle defined earlier by Holmes coincides with the extension of the double ramification cycle defined earlier by Kass and Pagani.
In the paper Ringel duality for certain strongly quasihereditary algebras, Martin Kalck and Joseph Karmazyn introduce quasihereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure, and present a uniform formula describing the Ringel duals of these quasihereditary algebras.
In the paper Homological mirror symmetry, coisotropic branes and \(P=W\), Ludmil Katzarkov and Leonardo Soriani discuss a possible approach to the famous conjecture of de Cataldo, Hausel and Migliorini via the theory of coisotropic brains. For Fano manifolds and their Landau–Ginsburg models, this conjecture is known as Katzarkov–Kontsevich–Pantev conjecture.
In the paper Cylinders in rational surfaces, Lisa Marquand and Joonyeong Won prove the following
Theorem
Let S be a smooth rational surface such that \(K_S^2\geqslant 3\), and let A be an ample \(\mathbb {Q}\)divisor on S. Then there exists an Apolar cylinder in S except the case when S is a smooth cubic surface and \(A\in \mathbb {Q}_{>0}[K_S]\).
 (C)

Open image in new window for some affine curve Z,
 (P)

there is an effective \(\mathbb {Q}\)divisor D on S such that \(D\sim _{\mathbb {Q}} A\) and Open image in new window .
Theorem
(Cheltsov, Kishimoto, Park, Prokhorov, Won, Zaidenberg) Let S be a smooth del Pezzo surface such that \(K_S^2\geqslant 3\), and let A be an ample \(\mathbb {Q}\)divisor on S. Then there exists an Apolar cylinder in S except the case when S is a smooth cubic surface and \(A\in \mathbb {Q}_{>0}[K_S]\).
Note also that if S is a smooth cubic surface and \(A\in \mathbb {Q}_{>0}[K_S]\), then S does not contain Apolar cylinders by a result of Cheltsov, Park and Won.
Theorem
Let V be a smooth Fano–Mukai fourfold in Open image in new window of degree 18. Then V contains a hyperplane section X such that X is singular along a cone over a rational twisted cubic curve, and Open image in new window .
In the paper Edge and Fano on nets of quadrics, Alessandro Verra revisits and reconstructs classical results of Edge and Fano about the family of scrolls of degree 8 in the complex projective space, whose plane sections are projected bicanonical models of a genus 3 curve. Verra shows that this beautiful subject is related to the moduli of semistable rank two vector bundles on genus 3 curves with bicanonical determinants.
In the paper Equivariant birational geometry of quintic del Pezzo surface, Jonas Wolter (explicitly) describes all Gequivariant birational transformations of the (unique) smooth del Pezzo surface of degree 5 into GMori fibre spaces, where Open image in new window . In particular, he proves that there are exactly two such GMori fibre spaces: Open image in new window and the del Pezzo surface of degree 5 itself. In particular, the smooth del Pezzo surface of degree 5 is not Gbirational to a conic bundle, and it is not Gbirational to Open image in new window .
The second part of Edge Volume is concluded by a postface William Leonard Edge by James Hirschfeld, who was the only research student of William Edge. James opened the first Edge Days back in 2013. His short note contains some personal memories about Edge.
William Edge stayed research active throughout his entire life. His very last paper 28 real bitangents has been published in 1994 when he was 90 years old. Geometry gave him energy, kept his spirits high and probably prolonged his life. We hope that Edge Volume will help to keep Edge’s legacy alive. The diversity of its contributions reflects the vitality of algebraic geometry in the directions impulsed by William Edge. Their authors have the same passion about mathematics as Edge had.
Fedor Bogomolov
Ivan Cheltsov