On the 17th of December 2016, twenty four mathematicians arrived to Easter Island. Alvaro Liendo and Sukhendu Mehrotra came from Chile, Sione Ma‘u came from Auckland in New Zealand, and four mathematicians came from Japan: Will Donovan, Kento Fujita, Yoshinori Gongyo, and Yujiro Kawamata. Five mathematicians flew from the States: Valery Alexeev came from Athens in Georgia, Fedor Bogomolov came from New York, Joe Cutrone came from Baltimore, Alexander Duncan came from Columbia in South Carolina, and Joaquin Moraga came from Salt Lake City. Nikita Kalinin and Ernesto Lupercio both came from Mexico City, DongSeon Hwang and Dima Sakovics came from South Korea, and the remaining eight mathematicians traveled from Europe: Elena Bunkova (Moscow), Ivan Cheltsov (Edinburgh), Lucas das Dores (Liverpool), Adrien Dubouloz (Dijon), Ilia Itenberg (Paris), Angelo Lopez (Rome), Grigory Mikhalkin (Geneva), and Navid Nabijou (London). All of them came to the island to participate in a conference on Algebraic Geometry. This volume carries 26 papers related to this conference.
Why Easter Island? We wanted to show that people can do Mathematics everywhere including the most remote places. Easter Island fits this goal perfectly, since it is one of the world’s most isolated inhabited islands: the nearest continental point is 3512 kilometers away in Chile; the closest inhabited islands are three Juan Fernández Islands (1850 kilometers to the east) and Pitcairn Island (2075 kilometers to the west). The only island that competes with Easter Island for the title of being the most remote is Tristan da Cunha in the southern Atlantic, which does not have an airport and can only be reached by sea.
Geometrically, Easter Island is a triangle whose sides are 16 kilometers, 18 kilometers, and 24 kilometers. Its area is approximately 164 square kilometers, and its longest side is the southeast coast of the island. Close to the vertices of this triangle, there are three extinct volcanoes: Rano Kau, Rano Raraku, and Rano Aroi. The highest point of Easter Island is the summit of Terevaka, which is 507 meters above the sea level. The island is very small and flat: it is possible to walk around the island in one day.
The history of Easter Island is interesting, and can be even called rich for such a lonely remote small place. It was first settled by a small group of Polynesians about or shortly after 400 CE. At this time, the island was covered by great palm forest. In the years following the initial settlement, islanders were cutting down trees, making canoes, and catching fish. A small initial population of Easter Island grew to a developed society, whose members built thousands of huge statues, called Moai, which was an enormous achievement for island’s inhabitants. This mega project played a gloomy role in island’s history: the islanders crucially overused their resources, so that the large trees completely disappeared from the island. As a result, the islanders were unable to build boats, and the loss of forests also led to soil erosion, causing lack of fresh water, which severely damages agriculture. During long stagnation period, islanders ate all birds and dogs, and catched all fish that lived close to the island, so that the only available food left were chicken, bananas, sweet potatoes, sugar cane and rats. Then, on Easter Sunday in 1722, the island was discovered by the Dutch explorer Jacob Roggeveen, who gave the island its name.
von Kotzebue was right: in 1805, a group of Americans arrived to Easter Island and took 12 hostages after a bloody fight. Later a series of similar devastating events killed or removed most of the population of Easter Island. The worst of them happened in 1862, when Peruvian slave raiders violently abducted around 1500 men and women. Later 15 survivors returned to Easter Island from Peru only to bring smallpox disease with them, which created a devastating epidemic.“The mistrustful attitude of islanders made me think that they probably had a quarrel with Europeans at some point, and Europeans destroyed their statues in revenge. I was also surprised that I did not see any women in the water or on the coast, while previous visitors complained about their bothering behavior. This convinced me that of course Europeans recently did here horrible things.”
In 1868, French mariner JeanBaptiste DutrouBornier, the agent of Tahiti based company the Maison Brander, began buying up land from islanders with legal help from Catholic mission. Eventually, he bought up all of the island apart from a small piece of land in around Hanga Roa, and turned Easter Island in one big sheep ranch. Islanders, completely deprived of their land, had to live in Hanga Roa as prisoners. Since the ranch did not need many people to take care of the sheep, DutrouBornier moved hundreds of islanders to Mangareva and Tahiti: only 111 of them remained on the island in 1877.
On the 9th of September 1888, Chile annexed Easter Island. But this did not help its native people: Chile leased the island to the Williamson–Balfour Company, which continued to run it as a sheep ranch until 1953 (the company constructed a wall around Hanga Roa to keep local people in the village). Between 1953 and 1966, the island was managed by the Chilean Navy. In 1966, all islanders were granted Chilean citizenship and became able to move freely in the island. In 2007, Easter Island gained special constitutional status that brought islanders peace and prosperity. Now, with a single exception of posh Hangaroa Eco Village Spa which hosted our conference, all land and local businesses on the island belong to the natives. Since tourism thrives, all islanders have very stable income that allows them to have a middle class life style, which was simply unimaginable few generations ago.
Relaxed life style and spiritual happiness of the islanders created very friendly atmosphere during our meeting, which helped to organize this complete volume: every participant submitted a paper. Since the collapse of Easter Island’s economics serves as a standard model in many problems on population dynamics, we also included three papers that describe the ecosystem of Easter Island using differential equations. In total, fifty one mathematicians contributed 26 papers to this volume. Let us describe their contribution.
Let X be a scheme of finite type over an arbitrary field. To measure the minimal homological complexity of the category \(\mathrm {D}^b(\mathrm {coh}(X))\), Raphaël Rouquier introduced a new invariant and proved that it is always at least the Krull dimension of the scheme X (with equality in certain cases). This invariant is now called the Rouquier dimension of the scheme X. Recently Dima Orlov posed the following
Conjecture
Suppose that X is smooth and projective. Then the Rouquier dimension of the scheme X is equal to the Krull dimension of the scheme X.
Orlov also proved this conjecture in dimension 1 (for smooth projective curves). In higher dimensions, Orlov’s conjecture seems to be very difficult to address, and all supporting evidence for it comes from individual constructions specialized to particular examples. In the paper The toric Frobenius morphism and a conjecture of Orlov, Matthew Ballard, Alexander Duncan, and Patrick McFaddin expand the set of varieties satisfying Orlov’s conjecture. To do this, they combine the Bondal–Uehara method for producing exceptional collections on toric varieties with an earlier result of Ballard and Favero. Moreover, Ballard, Duncan, and McFaddin prove Orlov’s conjecture for a (smooth and projective) scheme X in the case when the Bondal–Uehara method produces a tilting bundle on it.
As we already mentioned above, the deforestation of Easter Island led to a collapse of its ecosystem and population numbers. In the paper Ecological collapse of Easter Island and the role of price fixing, William and Wesley Basener review some of the factors that have been studied which may have contributed to the collapse. They show that the function for the harvesting rate is an important factor in dynamics that can lead to total collapse of the resource. In particular, they prove that harvesting rate functions resulting in total collapse correspond to fixed prices of the resources.
In the paper Homomorphisms of multiplicative groups of fields preserving algebraic dependence, Fedor Bogomolov, Marat Rovinsky, and Yuri Tschinkel study homomorphisms of multiplicative groups of fields preserving algebraic dependence and show that such homomorphisms give rise to valuations.
In the paper On the problem of differentiation of hyperelliptic functions, Elena Bunkova describes a construction that leads to an explicit solution of the problem of differentiation of hyperelliptic functions. Namely, she gives explicit generators for the Lie algebra Open image in new window of derivations of the fields of hyperelliptic functions Open image in new window of genus g for \(g\in \{1,2,3\}\).
In the paper Fujita decomposition over higher dimensional base, Fabrizio Catanese and Yujiro Kawamata generalize a result of Takao Fujita, on the decomposition of Hodge bundles over curves, to the case of a higher dimensional base.
Let X be a smooth threefold such that \(K_X\) is numerically effective (nef) and \(K_X^3>0\) (big). Then X is said to be a weak Fano threefold. Weak Fano threefolds with Picard rank 2 arise naturally in the study of birational maps between smooth Fano threefolds with Picard rank 1, which leads to the (old) problem of their classification. This classification, started by Jahnke, Peternell, and Radloff long time ago, is based on the classical approach to birational geometry of Fano threefolds originated in the works of Vasily Iskovskikh and Kiyohiko Takeuchi. Its first (and probably most important) step is to create a list of numerical possibilities for extremal contraction from X. Relatively recently, Joseph Cutrone and Nicholas Marshburn completed this numerical classification and provided huge tables that capture geometric realizations of already existing weak Fano threefolds. Cutrone and Marshburn also gave geometric constructions for many new numerical examples. Later, more geometric constructions were given by them and Maxim Arap, Jeremy Blanc, Ivan Cheltsov, Stephane Lamy, and Constantin Shramov, which filled almost all existence cells in Cutrone–Marshburn tables. However, the actual geometric existence of some cases is an open problem. In the paper A weak Fano threefold arising as a blowup of a curve of genus 5 and degree 8 on\({\mathbb {P}}^3\), Joseph Cutrone, Michael Limarzi, and Nicholas Marshburn sort out one such case: they construct a smooth weak Fano threefold of Picard number two with small anticanonical morphism that arises as a blowup of a smooth curve of genus 5 and degree 8 in \({\mathbb {P}}^3\).
In the paper Perverse schobers on Riemann surfaces: constructions and examples, Will Donovan studies perverse sheaves of categories on Riemann surfaces. Among other results, he constructs, for certain wall crossings in geometric invariant theory, a schober on the complex plane, singular at each imaginary integer. He also suggests an application to mirror symmetry.
Definition
Let \((S,\rho )\) be a minimal Gsurface such that S is a smooth Del Pezzo surface, and \(\mathrm{Pic}^G(S)\cong {\mathbb {Z}}\). Then \((S,\rho )\) is said to be Gbirationally rigid (Gbirationally superrigid, respectively) if there is no Gbirational map from S to any other minimal Gsurface (and the group of Gbiregular automorphisms of the surface S coincides with the group of its Gbirational automorphisms, respectively).
If \((S,\rho )\) is a Gsurface such that S is a smooth del Pezzo surface, \(K_S^2\leqslant 3\) and \(\mathrm{Pic}^G(S)\simeq {\mathbb {Z}}\), then \((S,\rho )\) is Gbirationally rigid by the classical theorem of Beniamino Segre and Yuri Manin. In the paper Gbirational superrigidity of Del Pezzo surfaces of degree 2 and 3, Lucas das Dores and Mirko Mauri refine this result by determining which Gsurfaces in the Segre–Manin theorem are Gbirationally superrigid and which are not.
In the paper The effect of tree diffusion in a twodimensional continuous model for Easter Island, István Faragó, Róbert Horváth, and Bálint Takács consider a twodimensional continuous model that describes the ecology of Easter Island. They show that the increase of the parameter corresponding to the diffusion of trees on the island has a stabilizing effect on the system, potentially preventing the collapse of island’s ecology.
Conjecture
In the paper The generalized Mukai conjecture for toric log Fano pairs, Kento Fujita proves this conjecture in the case when X is toric, and \(\Delta \) is a torus invariant \({\mathbb {Q}}\)divisor.
Let us remind the reader that Kollár’s injectivity theorem for semiample line bundles is the following classical result:
Theorem
In the paper Kollár’s type injectivity theorem for globallyFregular varieties, Yoshinori Gongyo and Shunsuke Takagi prove the following analogue of this theorem for varieties defined over a field of positive characteristic.
Theorem
In the paper Nearly Frobenius algebras, Ana González, Ernesto Lupercio, Carlos Segovia, and Bernardo Uribe study nearly Frobenius algebras, generalizations of Frobenius algebras which appear naturally in topology. They present foundational results about nearly Frobenius algebras and consider some applications in geometry, topology, and representation theory.
In the paper A curvedetecting formula for projective surfaces, DongSeon Hwang presents an intersectiontheoretic formula about curves on projective surfaces in terms of lattices with special emphasis on minimal resolutions of \({\mathbb {Q}}\)homology projective planes. This formula can be used to detect the existence or nonexistence of curves with given intersection properties.
In the paper Ordinary differential equations and Easter Island, Lorelei Koss gives a survey of ordinary differential equation models investigating environmental and sustainability issues in the history of Easter Island.
A \(\mathbb {T}\)variety is an algebraic variety endowed with an effective action of an algebraic torus. In the paper The fundamental group of log terminal\({\mathbb {T}}\)variety, Antonio Laface, Alvaro Liendo, and Joaquin Moraga study the fundamental group of varieties with log terminal singularities endowed with an algebraic torus \({\mathbb {T}}\)action. In particular, they prove the simple connectedness of the spectrum of the Cox ring of a complex Fano variety, and compute the fundamental group of rational log terminal \({\mathbb {T}}\)varieties of complexity one.

X is a surface with maximal Albanese dimension;

H is a very ample divisor on X such that Open image in new window .
In the paper Relating transfinite diameters using an Okounkov body, Sione Ma‘u studies some notions of pluripotential theory using an Okounkov body.
In the paper Examples of tropicaltoLagrangian correspondence, Grisha Mikhalkin associates Lagrangian submanifolds in symplectic toric varieties to certain tropical curves inside the convex polyhedral domains of \({\mathbb {R}}^n\) that appear as the images of the moment map of the toric varieties. In particular, for the case \(n=2\), Grisha reproves Givental’s theorem on Lagrangian embeddability of nonoriented surfaces to \({\mathbb {C}}^2\).
Let X be a smooth projective variety. Givental’s Lagrangian cone Open image in new window is a Lagrangian submanifold of a symplectic vector space which encodes the genuszero Gromov–Witten invariants of the variety X. Building on work of Sasha Braverman, Tom Coates obtained the Lagrangian cone as the pushforward of a certain class on the moduli space of stable maps to Open image in new window . In the paper The fundamental solution matrix and relative stable maps, Navid Nabijou recasts this construction in its natural context, namely the moduli space of stable maps to Open image in new window relative the divisor Open image in new window . He finds that the resulting pushforward is another familiar object, namely the transform of the Lagrangian cone under the action of the fundamental solution matrix. Nabijou also uses a hidden polynomiality property implied by his construction to obtain a sequence of universal relations for the Gromov–Witten invariants.

the plane \({\mathbb {P}}^2\) does not contain Gfixed points;

the group G is isomorphic neither to \({\mathfrak {A}}_4\) nor to \({\mathfrak {S}}_4\).
Notes
Acknowledgements
On behalf of the participants of this conference on Algebraic Geometry, we thank Universidad de Talca for its kind support.