MAGADAN VOLUME Fedor Bogomolov, Ivan Cheltsov, Tatiana Kozlovskaya, and Jihun Park On the 5th of December 2015, seventeen mathematicians arrived in Magadan, a small city in the Russian Far East, located on the Sea of Okhotsk between Nagaev and Gertner bays. Ten mathematicians flew in from Moscow: Elena Bunkova, Sergey Galkin, Sergey Gorchinskiy, Igor Netay, Yuri Prokhorov, Vic- tor Przyjalkowski, Marat Rovinsky, Constantin Shramov, Evgeny Smirnov and Andrey Trepalin. Andrey Mironov came from Novosibisrk, and Alexei Savva- teev from Irkutsk. The remaining five traveled from outside of Russia: Fedor Bogomolov came from New York in the US, Ivan Cheltsov from Edinburgh in the UK, Kento Fujita from Kyoto in Japan, Jihun Park and Dmitrijs Sakovics from Pohang in Korea. They all came to Magadan to participate in a con- ference. Two local mathematicians, Tatiana Kozlovskaya and Olga Starikova, joined them in Magadan. This volume carries 21 papers related to the Magadan Conference that was held at North{East State University in Magadan for the period 6{12 December 2015. Why Magadan? The idea of organizing a conference there came from the song \I am going to Magadan" by Vasya Oblomov. Ivan Cheltsov's kids (Vanya, Fedya and Katya) often listened to this song during the family trip to Bormes- les-Mimosas in August 2014. This song is not the only song about Magadan and songs about the city are somewhat surprisingly numerous. Nearly everyone in Russia knows that Magadan is the capital of Kolyma (these words are from another song \Port Vanino"). Unfortunately, the image of Magadan conjured up by songs, books and movies is usually somewhat less than enticing, to say the least. A typical example is the well-known dialogue in the comedy The Diamond Arm by Leonid Gaidai: | Come visit us in Kolyma! | No, I think it's better you visit us instead. Most Russians hearing the words \Magadan" or \Kolyma" would probably tend to immediately think of the land of eternal frost, labor camps and the Gulag. Not exactly your first choice if you were looking for a holiday destination. (Kento Fujita, one of the participants of the conference, visited Russia for the first time in his life. His first town in Russia was Magadan!) However, contemporary Magadan has managed to move on somewhat from the past, although the locals have not in any way been subsumed by a collective amnesia and forgotten the uncomfortable events of its history. In case anyone might, however, the Mask of Sorrow, a monument standing high on a hill above Magadan, serves as a reminder of the thousands of people who perished there. Magadan was founded in 1930 as a port of entry to Soviet arctic gold mines and also as the headquarters of the network of the Gulag camps dispersed i MAGADAN VOLUME throughout the huge Kolyma region. Facing south it also played a role as a resort town for the Kolyma which runs down from frozen mountains towards the Arctic Ocean. From 1932 to 1953, the city was operated by the Dalstroy organization that mined gold throughout the Kolyma region using extensively the work of prisoners living in the concentration camps. This company also built the famous Kolyma highway, a 2000 km road that connects Magadan to Yakutsk, the capital of Russian diamond production. This is the only road that links Magadan to the rest of the world. However, most people arrive either by air or by sea (no railways go to Magadan). Because of this, locals often feel like they live on an island and refer to other parts of Russia as the mainland. Despite its remote location, harsh climate (the daily mean temperature in December is −15◦C), small size (less than 100,000 people) and turbulent history, Magadan is a vibrant modern city. In fact, the hardships tend to bring people together and locals are almost always ready to help a stranger. Interestingly, many people who thought they had left Magadan forever returned later, drawn back by Magadan's nature and the openness of its people. The economy is mostly focused on the gold mining and machine-building industries. The city hosts the headquarters of all local gold mining companies, but the gold is not mined or refined there. Because of this, Magadan is a middle class city. It has a number of cultural institutions including a a very rich anthropological museum, a somewhat unique kind of geological museum, and an extremely informative museum of the Gulag history. It also has a big library, several modern cinemas and a famous drama theater. In addition to this, Magadan is a home to North{East State University, which has around three thousand students. The university provides excellent education for locals, and has all possible facilities for an international conference. We are very grateful to the university for hosting our Magadan Conference. Nineteen participants is not very many for a scientific meeting. In fact, we expected more people to join the conference. Unfortunately, Transaero, the only airline that operated the direct Moscow to Magadan flight, went into liquidation on the 1st of October 2015. Because of this many Europeans and Americans canceled their participation. Despite this, the Magadan Conference was a successful event. Thirty mathematicians (those who came to Magadan and those who planned to come but failed to do so) contributed 21 papers to this volume. Briefly their contribution are as follows. In the paper Endomorphisms of projective bundles over a certain class of varieties, Ekaterina Amerik and Alexandra Kuznetsova study endomorphisms of projectivizations of vector bundles on simply-connected projective varieties. To be precise, let X be a simply-connected projective variety such that the first cohomology groups of all line bundles on X are zero, and let E be a vector bundle over X. Then a power of any endomorphism of P(E) must map fibers of the natural projection P(E) ! X back to fibers. This is not very hard to show. In their paper, Amerik and Kuznetsova proved that if E admits an endomorphism which is of degree greater than 1 on the fibers, then E splits into a direct sum of line bundles. ii F. BOGOMOLOV, I. CHELTSOV, T. KOZLOVSKAYA, AND J. PARK In the paper Equivariant matrix factorizations and Hamiltonian reduction, Sergey Arkhipov and Tina Kanstrup established an equivalence of two categories related to the moment map ∗ ∗ µ: TX ! g : Here X is a smooth scheme with an action of an algebraic group G, and g is the Lie algebra of the group G. The first category is the derived category of G-equivariant coherent sheaves on the derived fiber µ−1(0), and the second ∗ category is the derived category of G-equivariant matrix factorizations on TX ×g with potential given by µ. To describe the paper Cylinders in del Pezzo surfaces with Du Val singu- larities by Grigory Belousov, let us fix a del Pezzo surface Sd with Du Val singularities such that K2 = d. For 3 d 9, the anticanonical divisor is very Sd 6 6 ample and the anticanonical linear system embeds Sd into the projective space d 3 P . In particular, for d = 3, the surface S3 is a cubic surface in P . Meanwhile, for d = 2, the surface Sd is a hypersurface in the weighted projective space P(1; 1; 1; 2) defined by a quasi-homogeneous polynomial of degree 4. Similarly, for d = 1, the surface Sd is a hypersurface P(1; 1; 2; 3) of degree 6. Definition. For an ample Q-divisor H on the surface Sd, an H-polar cylinder in Sd is an open subset U ⊂ Sd such that • the subset U is isomorphic to Z × A1 for some affine curve Z, • there exists an effective Q-divisor D on the surface Sd such that U = S n Supp(D) and D ∼Q −KSd . The group of Kishimoto, Prokhorov, and Zaidenberg and the group of Cheltsov, Park, and Won have studied the existence of (−KSd )-polar cylinders in the surface Sd. In the case when Sd is smooth, they showed that Sd has a (−KSd )- polar cylinder if and only if d > 4. In the singular case, Cheltsov, Park and Won proved the following. Theorem. Suppose that the surface Sd is singular. Then it does not contain (−KSd )-polar cylinders if and only if • either d = 1 and Sd allows only singular points of types A1, A2, A3 or D4, • or d = 2 and Sd allows only singular points of type A1. Moreover, they pointed out that the existence of (−KSd )-polar cylinders is closely related to the existence of non-log canonical tigers on the surfaces Sd. The latter notion was introduced by McKernan and Keel. Let us recall that a tiger on Sd is an effective Q-divisor D such that D ∼Q −KSd ; and the singularities of the log pair (Sd;D) are not Kawamata log terminal. If in addition, the singularities of the log pair (Sd;D) are not log canonical, then D is said to be non-log canonical tiger. The surface Sd contains a non-log canonical tiger if and only if α(Sd) < 1, where α(Sd) is the α-invariant of Tian of the surface Sd. The number α(Sd) is iii MAGADAN VOLUME computed in all possible cases, and in most of them we have α(Sd) < 1. For 5 example, if d > 2, then α(Sd) 6 6 . Thus, most del Pezzo surfaces with Du Val singularities contain non-log canonical tigers. However, many such tigers are just divisors in the anticanonical linear system j − KSd j, or effective divisors whose support contains tigers in j − KSd j.