Authors:Ciro Ciliberto; Th. Dedieu; F. Flamini; R. Pardini; C. Galati; S. Rollenske Pages: 5 - 11 Abstract: A problem session, has been held during the workshop “Birational geometry of surfaces” which took place at the Department of Mathematics of the University of Rome “Tor Vergata”, in January, 11–15, 2016. In the following paper, we gather problems and questions that have been proposed and discussed during the event. PubDate: 2018-03-01 DOI: 10.1007/s40574-018-0158-0 Issue No:Vol. 11, No. 1 (2018)

Authors:Francesco Bastianelli Pages: 13 - 25 Abstract: This survey retraces the author’s talk at the Workshop Birational geometry of surfaces, Rome, January 11–15, 2016. We consider various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given projective variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational. Then we review a series of results describing these invariants for various classes of projective surfaces. PubDate: 2018-03-01 DOI: 10.1007/s40574-017-0116-2 Issue No:Vol. 11, No. 1 (2018)

Authors:Francesco Bastianelli; Ciro Ciliberto; Flaminio Flamini; Paola Supino Pages: 31 - 38 Abstract: This short paper concerns the existence of curves with low gonality on smooth hypersurfaces \(X\subset \mathbb {P}^{n+1}\) . After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if \(X\subset \mathbb {P}^{n+1}\) is a very general hypersurface of degree \(d\geqslant 2n+2\) , the least gonality of a curve \(C\subset X\) passing through a general point of X is \(\mathrm {gon}(C)=d-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor \) , apart from some exceptions we list. PubDate: 2018-03-01 DOI: 10.1007/s40574-017-0129-x Issue No:Vol. 11, No. 1 (2018)

Authors:Julie Decaup; Adrien Dubouloz Pages: 39 - 54 Abstract: We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular automorphisms outside Q, there are exactly two such lines: the restriction of a smooth conic osculating Q at a rational point and the restriction of the tangent line to Q at a rational point. In contrast, we give examples illustrating the fact that over fields of positive characteristic, there exist exotic closed embeddings of the affine line in the complement of Q. We also determine an explicit set of birational endomorphisms of the plane whose restrictions generates the automorphism group of the complement of Q over a field of arbitrary characteristic. PubDate: 2018-03-01 DOI: 10.1007/s40574-017-0119-z Issue No:Vol. 11, No. 1 (2018)

Authors:Filippo F. Favale Pages: 55 - 67 Abstract: In this paper we are interested in quotients of Calabi–Yau threefolds with isolated singularities. In particular, we analyze the case when X / G has terminal singularities. We prove that, if G is cyclic of prime order and X / G has terminal singularities, then G has order lower than or equal to 5. PubDate: 2018-03-01 DOI: 10.1007/s40574-017-0128-y Issue No:Vol. 11, No. 1 (2018)

Authors:Claudio Fontanari; Diletta Martinelli Pages: 69 - 74 Abstract: In this survey we borrow from Coskun and Huizenga an example of application of Bridgeland stability conditions to birational geometry and we rephrase it without assuming any previous knowledge about derived categories. PubDate: 2018-03-01 DOI: 10.1007/s40574-016-0091-z Issue No:Vol. 11, No. 1 (2018)

Authors:Marco Franciosi; Sönke Rollenske Pages: 75 - 91 Abstract: Extending the description of canonical rings from Reid (J Fac Sci Univ Tokyo Sect IA Math 25(1):75–92, 1978) we show that every Gorenstein stable Godeaux surface with torsion of order at least 3 is smoothable. PubDate: 2018-03-01 DOI: 10.1007/s40574-016-0114-9 Issue No:Vol. 11, No. 1 (2018)

Authors:Margherita Lelli-Chiesa Pages: 93 - 105 Abstract: We survey some results concerning Severi varieties and variation in moduli of curves lying on K3 surfaces or on abelian surfaces. A number of open problems is listed and some work in progress is mentioned. PubDate: 2018-03-01 DOI: 10.1007/s40574-017-0126-0 Issue No:Vol. 11, No. 1 (2018)

Authors:Francesco Polizzi Pages: 107 - 119 Abstract: We relate the existence of some surfaces of general type and maximal Albanese dimension to the existence of some monodromy representations of the braid group \(\mathsf {B}_2(C_2)\) in the symmetric group \(\mathsf {S}_n\) . Furthermore, we compute the number of such representations up to \(n=9\) , and we analyze the cases \(n \in \{2, \, 3, \, 4\}\) . For \(n=2, \, 3\) we recover some surfaces with \(p_g=q=2\) recently studied (with different methods) by the author and his collaborators, whereas for \(n=4\) we obtain some conjecturally new examples. PubDate: 2018-03-01 DOI: 10.1007/s40574-017-0131-3 Issue No:Vol. 11, No. 1 (2018)

Authors:Sofia Tirabassi Pages: 121 - 124 Abstract: We show that the (twisted) derived category “recognizes” the three different kinds of Enriques surfaces in characteristic 2. PubDate: 2018-03-01 DOI: 10.1007/s40574-016-0100-2 Issue No:Vol. 11, No. 1 (2018)

Authors:Lucio Boccardo Abstract: In this paper we study two semilinear Dirichlet problems; the linear parts (in some sense, in duality) are a problem with singular convection term and a problem with singular drift. The nonlinear lower order terms have a regularizing effect: the solutions of the corresponding linear problems are less regular. PubDate: 2018-04-05 DOI: 10.1007/s40574-018-0160-6

Authors:Matteo Longo; Stefano Vigni Abstract: We extend to the supersingular case the \(\Lambda \) -adic Euler system method (where \(\Lambda \) is a suitable Iwasawa algebra) for Heegner points on elliptic curves that was originally developed by Bertolini in the ordinary setting. In particular, given an elliptic curve E over \(\mathbb {Q}\) with supersingular reduction at a prime \(p\ge 5\) , we prove results on the \(\Lambda \) -corank of certain plus/minus p-primary Selmer groups à la Kobayashi of E over the anticyclotomic \(\mathbb {Z}_p\) -extension of an imaginary quadratic field and on the asymptotic behaviour of p-primary Selmer groups of E when the base field varies over the finite layers of such a \(\mathbb {Z}_p\) -extension. These theorems can be alternatively obtained by combining results of Nekovář, Vatsal and Iovita–Pollack, but do not seem to be directly available in the current literature. PubDate: 2018-02-10 DOI: 10.1007/s40574-018-0162-4

Authors:Luisa Consiglieri Abstract: This paper deals with thermoelectric problems including the Peltier and Seebeck effects. The coupled elliptic and doubly quasilinear parabolic equations for the electric and heat currents are stated, respectively, under power-type boundary conditions that describe the thermal radiative effects. To verify the existence of weak solutions to this coupled problem (Theorem 1), analytical investigations for abstract multi-quasilinear elliptic-parabolic systems with nonsmooth data are presented (Theorems 2 and 3). They are essentially approximated solutions based on the Rothe method. It consists on introducing time discretized problems, establishing their existence, and then passing to the limit as the time step goes to zero. The proof of the existence of time discretized solutions relies on fixed point and compactness arguments. In this study, we establish quantitative estimates to clarify the smallness conditions. PubDate: 2018-01-18 DOI: 10.1007/s40574-018-0159-z

Authors:Aibo Liu; Changchun Liu Abstract: We consider a simplified smectic-A liquid crystal model in two dimensional bounded domain. The model consists of a Navier–Stokes equation governing the fluid velocity coupled with the transported heat flows of harmonic map. Based on the combination of the suitable dissipative estimates with the energy techniques, we established the existence of global attractor \(\mathcal {A}\) on a suitable phase-space and proved that the system has a regular compact absorbing set. PubDate: 2018-01-15 DOI: 10.1007/s40574-018-0156-2

Authors:Kathryn E. Hare; Jimmy He Abstract: Let G be a compact, connected simple Lie group and \(\mathfrak {g}\) its Lie algebra. It is known that if \(\mu \) is any G-invariant measure supported on an adjoint orbit in \(\mathfrak {g}\) , then for each integer k, the k-fold convolution product of \(\mu \) with itself is either singular or in \( L^{2}\) . This was originally proven by computations that depended on the Lie type of \(\mathfrak {g}\) , as well as properties of the measure. In this note, we observe that the validity of this dichotomy is a direct consequence of the Duistermaat–Heckman theorem from symplectic geometry and that, in fact, any convolution product of (even distinct) orbital measures is either singular or in \(L^{2+\varepsilon }\) for some \(\varepsilon >0\) . An abstract transference result is given to show that the \(L^{2}\) -singular dichotomy holds for certain of the G-invariant measures supported on conjugacy classes in G. PubDate: 2018-01-11 DOI: 10.1007/s40574-017-0154-9

Authors:Maurizio Brunetti; Adriana Ciampella Abstract: Let p be an odd prime. In this paper we determine the group of length-preserving automorphisms for the ordinary Steenrod algebra A(p) and for \({\mathcal {B}}(p)\) , the algebra of cohomology operations for the cohomology of cocommutative \(\mathbb {F}_p\) -Hopf algebras. Contrarily to the \(p=2\) case, no length-preserving strict monomorphism turns out to exist. PubDate: 2018-01-10 DOI: 10.1007/s40574-017-0155-8

Authors:Francesco Bastianelli Abstract: The purpose of this note is to fix an imprecision in Theorem 3.3 of the original paper, which affects the correctness of the statement and leads to a mistake PubDate: 2018-01-08 DOI: 10.1007/s40574-017-0153-x

Authors:Rasoul Akrami; Ali Reza Janfada; Mohammad Reza Miri Abstract: Let \(\mathcal {M}\) be a Hilbert \(C^*\) -module. Set \( x =\big <x,x\big >^\frac{1}{2}\) for every \(x\in \mathcal {M}\) and \(\mathcal {C}=\overline{\{\big <x,y\big >:x,y\in \mathcal {M}\}}\) . We prove that \( xa = x a \) for every \(x\in \mathcal {M}\) and \(a\in \mathcal {C}\) if and only if \(\mathcal {C}\) is a commutative \(C^*\) -algebra. PubDate: 2017-12-26 DOI: 10.1007/s40574-017-0152-y