Projective geometry and syzygies


Kiryong Chung (KIAS Seoul)

Title: On the geometry of the moduli space of pure sheaves supported on quartic space curves
Abstract: As a continuation of the work of Freiermuth and Trautmann, I study the geometry of the moduli space
of stable sheaves on P^3 with Hilbert polynomial $4m+1$.The moduli space has three irreducible components whose generic elements are, respectively,sheaves supported on rational quartic curves, on elliptic quartic curves, or on planar quartic curves.The main idea of the proof is to relate the moduli space with the Hilbert scheme of curves by wall crossing. I also present all stable sheaves contained in the intersections of the three irreducible components. This is joint work with J. Choi and M. Maican.

Slawomir Cynk (Jagiellonian University in Kraków),

Title: Picard-Fuchs operators of order four
Abstract: In the talk I will report on the long lasting search for Picard-Fuchs operators corresponding to one parameter families of Calabi-Yau threefolds. These operators are threedimensional counterparts of the hypergeometric operator satisfied by the period integral of elliptic curves. By mirror symmetry they are supposed to be related to the Gromov-Witten invariants of Calabi-Yau threefolds with Picard number one.

Kangjin Han (DGIST, Daegu),

Title: Syzygy bounds on the cubic strand of a projective variety and 3-linear resolutions
Abstract: Let X be any projective variety in P^N over an algebraically closed field K. Few years ago, K. Han and S. Kwak developed a technique to compare syzygies under projections, as applications they proved sharp upper bounds on the ranks of higher linear syzygies, and characterized the extremal and next-to-extremal cases. In this talk, we report generalizations of these results, which are done with S. Kwak and J. Ahn. We consider a basic degree bound and sharp bounds for generators and syzygies in this cubic strand. Further, the extremal cases will be discussed and open problems in this direction also introduced in the end.

Grzegorz Kapustka (IMPAN Warsaw),

Title: Twenty incident planes in P^5
Abstract: Answering to a problem of O’Grady we show a construction of a complete family of 20 incident planes in P^5. The construction is related to the geometry of the IHS fourfold constructed by Donten-Bury and Wiśniewski and with the Debarre-Varley abelian fourfold. This is a joint work in progress with M. Donten-Bury, M. Kapustka, B. van Geemen, J. Wiśniewski.

Sijong Kwak (KAIST Daejeon),

Title: A bound for Castelnuovo-Mumford regularity by double point divisors
Abstract. Let X be a non-degenerate smooth projective variety of degree d and codimension e. As motivated by the regularity conjecture due to Eisenbud and Goto, we first consider and revisit Mumford's method based on the geometric properties of double point divisors. By considering double point divisors from inner projection and its semi-positivity, we can show $reg(O_X) <= d-e$, and we classify the extremal and the next to extremal cases. Then, we also obtain a slightly better bound for Castelnuovo-Mumford regularity under suitable assumptions.

Jinhyung Park (KIAS Seoul),

Title: Syzygies of Cox rings of del Pezzo surfaces
Abstract: There has been considerable interest in giving explicit descriptions of Cox rings of del Pezzo surfaces. The main problem was the Batyrev-Popov conjecture which describes the relations, and Testa--Varilly-Alvarado--Velasco and Sturmfels--Xu independently verified this conjecture. It is now natural to study syzygies of relations of Cox rings of del Pezzo surfaces. I prove basic properties of Cox rings of del Pezzo surfaces and calculate some important invariants such as Hilbert functions, projective dimensions, and Castelnuovo-Mumford regularities. Finally, the graded betti numbers of Cox rings of some del Pezzo surfaces are determined. This is joint work in progress with Joonyeong Won.

Piotr Pragacz (IMPAN Warsaw)

Title: On diagonals of flag bundles
Abstract: We express the absolute diagonals of projective, Grassmann and, more generally, flag bundles using the zero schemes of some vector bundle sections. We do the same for the single point subschemes in flag bundles. We discuss diagonal and point properties related to flag bundles.

Sukmoon Huh (Sungkyunkwan University in Seoul),

Title : On connectedness of Hilbert scheme of locally CM curves on Segre threefolds
Abstract: Ever since Grothendieck showed the existence of Hilbert scheme in the 1960s, there have been many intensive works on this object. But surprisingly little is known about their individual structures. Especially the Hilbert schemes of curves with fixed degree and genus is proven to be connected by Hartshorne, while the connectedness of its open subset consisting of locally Cohen-Macaulay (for short, CM) curves, is known only for very small degree or for very large genus. In this talk we recall the notion of ribbon over projective lines and apply it to study the geometry of Hilbert schemes of locally CM curves with small degree on Segre threefold and discuss their connectedness. This is a joint work with E. Ballico.

Kristian Ranestad (University of Oslo)

Title: On the powersums presenting a general cubic form in 6 variables.
Abstract: A general cubic form in 6 variables is the sum of ten cubes of linear forms in a 4-dimensional variety of ways.  The linear forms involved form a hypersurface in the dual P^5.  Using ideas of von Bothmer I shall show how syzygies may be used to analyze this hypersurface.

Sławomir Rams (Leibniz University in Hannover),

Title: On lines on non-K3 quartic surfaces in three-dimensional projective space
Abstract: It is well known that every singular cubic surface that is not ruled by linescontains strictly less than 27 lines. In the case of complex quartic surfaces one can show that there more than 2000 possible configurations of singularities on non-K3 surfaces (A. Degtyarev). In my talk I will sketch a proof of  a bound on the number of lines on non-K3 quartic surfaces (joint work with  Víctor González Alonso/Hannover) and discuss various examples of singular quartic surfaces with many lines. Combined with earlier results on quartic surfaces this yields a sharp bound on the number of lines  on complex affine quartics and quartics with at most isolated singularities.

Francesco Russo (University of Catania)

Title: Every cubic four fold in $\mathcal C_14$ is rational.
Abstract: Let $\mathcal C$ be the moduli space of smooth cubic hypersurfaces in $\mathbb P^5$. The Fano-Hassett divisor $\mathcal C_14 \subseteq \mathcal C$ is defined as the closure of the locus of smooth cubic hypersurfaces in $\mathbb P^5$ containing a smooth rational normal scroll of degree 4.
The general member of $\mathcal C_14$ is known to be rational. We shall prove that every smooth cubic four fold in $\mathcal C_14$ is rational. We shall also present the naive parameter count pointing towards $\mathcal C_14=\mathcal C$ and Fano's beautiful geometrical proof that on the contrary $\mathcal C_14$ is a divisor.
If time allows we shall also outline the classification of cubic four folds containing a plane and some other small degree smooth surfaces, showing their relevance for some rationality questions.