Programme

Three main lecture series:

• Moduli spaces of algebraic surfaces --  Fabrizio Catanese (Univeristaet Bayreuth)

Plan:

• Classical topics on algebraic surfaces --  Ciro Ciliberto (University of Rome Tor Vergata),
• Bridgeland stability conditions -- Barbara Bolognese (University of Sheffield).

Research Talks:

1. DongSeon Hwang (Ajou University / University of Warwick),
2. Karol Palka (Polish Academy of Sciences),
3. Xavier Roulleau (Aix-Marseille Université),
4. Mike Roth (Queen's University Kingston Ontario),
5. Alessandra Sarti (Université de Poitiers),
6. Matthias Schuett (Leibniz Universitaet Hannover),
7. Giancarlo Urzua (Pontificia Universidad Católica de Chile),
8. Liana Heuberger (University of Warwick),
9. Roberto Laface (Technische Universität München),
10. Gebhard Martin (Technische Universität München),
11. Łukasz Sienkiewicz (University of Warsaw).

• 23Apr
  08:45 - 09:00 (Audytorium Danka)

  Welcome

  09:00 - 10:00 (Audytorium Danka)

  Ciro Ciliberto, Lecture I

  10:00 - 10:30 (Audytorium Danka)

  Coffee Break

  10:30 - 11:30 (Audytorium Danka)

  Ciro Ciliberto, Lecture II

  11:30 - 11:45

  Pause

  11:45 - 12:45 (Audytorium Danka)

  Fabrizio Catanese, Lecture I

  12:45 - 14:30

  Lunch Break

  14:30 - 15:15 (Audytorium Danka)

  Xavier Roulleau, Construction of Nikulin configurations on some Kummer surfaces

  Joint work with Alessandra Sarti.
  A Nikulin configuration on a K3 surface is a set C of 16 smooth disjoint rational curves. A famous result of Nikulin is that any K3 surface X containing a Nikulin configuration is a Kummer surface, which means that there exists an abelian surface A such that X is the minimal resolution of the quotient A/[-1] and the exceptional curves of the resolution X->A[-1] are the 16 curves of the Nikulin configuration C (this is denoted X=Km(A)).
  In this talk, starting with a Kummer configuration C on some polarised Kummer surface X, we will construct another Kummer configuration C’ on X such that if A and A’ denotes the associated Abelian surfaces, although one has Km(A)=X=Km(A’), the Abelian surfaces A and A’ are not isomorphic (unless X is a Jacobian Kummer surface). That construction uses the Torelli Theorem for K3. We obtain also some new knowledge on the automorphisms group of Kummer surfaces and it brings some new special configurations of rational curves on K3’s. If time permits, we will derive some applications to the construction of interesting surfaces of general type, like the Schoen surfaces.

  15:15 - 15:45

  Coffee Break

  15:45 - 16:30 (Audytorium Danka)

  Alessandra Sarti, Involutions on the Hilbert scheme of two points coming from special K3 surfaces.

  Abstract: I will show how to use K3 surfaces with special geometry to construct non-symplectic  involutions on their Hilbertscheme of two points.  More precisely, I will consider K3 surfaces with Picard number two that admit several embeddings as smooth quartics in the 3-dimensional comple x projective space and K3 surfaces with several nodes.I will then show that the induced involutions on the Hilbert scheme have strong connections to the famous Beauville involution.
  This is a joint work with S. Boissière and A. Cattaneo.

  16:30 - 16:45 (Audytorium Danka)

  Break

  16:45 - 17:30 (Audytorium Danka)

  Karol Palka, Coolidge-Nagata conjecture and generalizations.

  Abstract: In 2017 we proved, together with M. Koras, the Coolidge-Nagata conjecture, which asserts that every curve in the projective plane which is topologically a line is Cremona equivalent to a line. We will discuss more general conjectures and a general method of finding (-1)-curves on smooth surfaces behaving nicely with respect to a given Q-divisor, which was applied in the proof.

• 24Apr
  09:00 - 10:00 (Audytorium Danka)

  Ciro Ciliberto, Lecture III

  10:00 - 10:30

  Coffee Break

  10:30 - 11:30 (Audytorium Danka)

  Fabrizio Catanese, Lecture II

  11:30 - 11:45

  Pause

  11:45 - 12:45 (Audytorium Danka)

  Barbara Bolognese, Lecture I

  12:45 - 14:30

  Lunch Break

  14:30 - 15:15 (Audytorium Danka)

  Giancarlo Urzua, Optimal bounds for singularities in stable surfaces

  Kollár and Shepherd-Barron (1988) introduced a natural compactification to the Gieseker moduli space of surfaces of general type, which is analogous to the Deligne-Mumford (1969) compactification of the moduli space of curves of genus g>1. This compactification is coarsely represented by a projective scheme (due to Kollár 1990) because of Alexeev's proof of boundedness (1994). Surfaces parametrized by this projective scheme are called (KSBA) stable surfaces, which may or may not be degenerations of canonical projective surfaces of general type (i.e. at most ADE singularities and K ample). In this talk, we concentrate on stable surfaces which have at most cyclic quotient singularities. When these stable surfaces come from a degeneration, then the cyclic quotient singularities are T-singularities, this is, of the form 1/dn^2(1,dna-1) (with gcd(n,a)=1, n>1). After fixing K^2, we have a finite list of such singularities. It is a hard problem to write down that list.

  This talk is about a joint work with Julie Rana, where we explicitly bound T-singularities on stable surfaces W with one singularity. This bound depends on K^2, and it is optimal when W is not rational. We classify and realize surfaces attaining the bound for each Kodaira dimension of the minimal resolution of W. At the end, I will explain the situation when W is rational, which in particular leads to a problem of finding minimal Cremona degrees of plane curves.

  15:15 - 15:45

  Coffee Break

  15:45 - 16:15 (Audytorium Danka)

  Roberto Laface, Bounding negativity on blow-ups of algebraic surfaces

  The Bounded Negativity Conjecture (BNC) has a long oral tradition, and it seems to date back to F. Enriques​. In some cases, the conjecture is known to hold true, but the problem acquires a completely different flavor when we start considering non-minimal surfaces, for example blow-ups of those surfaces for which the BNC is true. In those cases, it still appears that not much can be said. In a joint project with Piotr Pokora, we gather evidence towards the validity of the Weighted BNC by constructing line bundles that yield a linear bound on the self-intersection of the curves.
  16:15 - 17:30 (Audytorium Danka)

  Poster Session

• 25Apr
  09:00 - 10:00 (Audytorium Danka)

  Barbara Bolognese, Lecture II

  10:00 - 10:30

  Coffee Break

  10:30 - 11:30 (Audytorium Danka)

  Fabrizio Catanese, Lecture III

  11:30 - 11:45

  Pause

  11:45 - 12:45 (Audytorium Danka)

  Ciro Ciliberto, Lecture IV

  12:45 - 14:30

  Lunch Break

  14:30 - 15:15 (Classroom 216 (Institute of Mathematics))

  DongSeon Hwang, On cascades of log del Pezzo surfaces

  Abstract: There have been numerous attempts to classify log del Pezzo surfaces. In this talk, I will quickly summarize the known results to this goal and report my work on the classification of such surfaces. The result is obtained by generalizing the notion of ‘cascades’ of nonsingular del Pezzo surfaces to the singular case. This approach is turned out to be successful when the surface is toric or of Picard number one. At the end of the talk, I will also speak on the Singular Surface Database(SSDB) project which aims to realize this method.
   

  15:15 - 15:30

  Pause

  15:30 - 16:00 (Classroom 216 (Institute of Mathematics))

  Łukasz Sienkiewicz, Vinberg's X_4 revisited

  During the talk I will discuss a unique complex K3 surface with maximal Picard rank and with discriminant equal to four. I will describe configuration of smooth, rational curves on this surface and identify generators of its automorphisms group with distinguished elements of the Cremona group of P^2 . This work may be considered as an extension of Vinberg’s research on the subject.

  18:30 - 21:00 (Restaurant Sukiennice (Cracow Main Square))

  Conference Dinner

• 26Apr
  09:00 - 10:00 (Audytorium Danka)

  Ciro Ciliberto, Lecture V

  10:00 - 10:30

  Coffee Break

  10:30 - 11:30 (Audytorium Danka)

  Fabrizio Catanese, Lecture IV

  11:30 - 11:45

  Pause

  11:45 - 12:45 (Audytorium Danka)

  Barbara Bolognese, Lecture III

  12:45 - 14:30

  Lunch Break

  14:30 - 15:15 (Audytorium Danka)

  Matthias Schuett, Zariski K3 surfaces

  It is a classical fact in algebraic geometry that unirational curves are rational, and the same holds true for surfaces in characteristic zero, but not in positive characteristic or in higher dimension.
  In this talk I will report on joint work with T. Katsura to construct Zariski K3 surfaces, i.e. K3 surfaces admitting a purely inseparable map of degree p from the projective plane. In particular, we will prove that any supersingular Kummer surface is Zariski in certain characteristics.
   

  15:15 - 15:45

  Coffee Break

  15:45 - 16:15 (Audytorium Danka)

  Gebhard Martin, Numerically trivial automorphisms of Enriques surfaces in positive characteristic

  
  The first examples of surfaces for which the action of the automorphism group on cohomology is not faithful, even though the group itself is discrete, were Enriques surfaces. In 1984, S. Mukai and Y. Namikawa obtained a complete classification of complex Enriques surfaces with such numerically trivial automorphisms. I will explain how to obtain the classification of possible numerically trivial automorphism groups of Enriques surfaces in arbitrary positive characteristic.
  This is joint work with I. Dolgachev.

• 27Apr
  09:00 - 10:00 (Audytorium Danka)

  Fabrizio Catanese, Lecture V

  10:00 - 10:30

  Coffee Break

  10:30 - 11:30 (Audytorium Danka)

  Barbara Bolognese, Lecture IV

  11:30 - 11:45

  Pause

  11:45 - 12:45 (Audytorium Danka)

  Barbara Bolognese, Lecture V

  12:45 - 14:30

  Lunch Break

  14:30 - 15:15 (Audytorium Danka)

  Mike Roth, r-point Seshadri constants on P¹ x P¹.

  The r-point Seshadri constant is a way of studying the nef and ample cones on the blow up of a variety at $r$ general points.  Many basic questions about this constant are unknown, including on P^2, where questions about the r-point Seshadri constant are equivalent to the Nagata conjecture.

  This talk will focus on the r-point Seshadri constants on P^1 x P^1.  One advantage of this surface is that there is an infinite order ‘Cremona autormorphism’ preserving the equimultiplicity subspace, and so allows one to take known information and iterate it.  This easily allows one to determine the nef and effective cones in the ‘K_X-negative zone’, and to produce irrational nef classes (although not ones which produce irrational Seshadri constants).

  15:15 - 15:30

  Pause

  15:30 - 16:00 (Audytorium Danka)

  Liana Heuberger, Fano fourfolds and K3 surfaces with small invariants'

  One of the more concrete ways of approaching mirror symmetry is to study Q-Gorenstein smoothings of singular toric Fano varieties. The polytopes associated to these varieties are usually reflexive, and for threefolds they help recover the famous classification of Mori and Mukai. 

   

  This approach, though very efficient, fails to find smooth Fanos with "small" invariants, for example del Pezzo surfaces of degrees one and two. In this talk, I will give a precise definition of "small" in dimension four, and use K3 surfaces contained inside these fourfolds to outline a method of classification. This is joint work in progress with Alessio Corti.
  16:00 - 17:00 (Audytorium Danka)

  Goodbye Coffee