# Polar geometry

A conference celebrating the work of Ragni Piene on the occasion of her 70th birthday.

The talks will start on Thursday January 26th at 11am in **Auditorium 2**, in the **Vilhelm Bjerknes building** at the University of Oslo. See the following link for details how to find this room: https://www.uio.no/om/finn-

Each day, there will be a small lunch served in the 12th floor of Niels Henrik Abels hus, which is the building next to the Vilhelm Bjerknes building.

We will also have a conference dinner on Friday January 27th at 7pm. This will take place at the Norwegian Academy of Science and Letters, which is located in the central part of Oslo. See http://english.dnva.no/ for directions.

### Program

**Thursday 26th**

11:00-11:10 | Opening |

11:10-12:00 | Kleiman |

12:20-13:10 | McDuff |

13:15-14:15 | Lunch |

14:30-15:20 | Mallavibarrena |

15:40-16:30 | Teissier |

**Friday 27th**

09:00-09:50 | Dickenstein |

10:15-11:05 | Vainsencher |

11:25-12:15 | Villamizar |

12:20-13:20 | Lunch |

13:30-14:20 | Postinghel |

14:40-15:30 | Göttsche |

Conference dinner at 19:00 at Det Norske Vitenskapsakademi.

**Saturday** **28th**

09:00-09:50 | Di Rocco |

10:15-11:05 | Rennemo |

11:25-12:15 | Aluffi |

12:20-13:20 | Lunch |

### Titles and Abstracts:

**Steven Kleiman**

*Ragni's Work in Polar Geometry*

Abstract: We'll discuss how Ragni came to work in Polar Geometry, how Polar Geometry arose between 1805 and 1839, what Ragni did in her 1976 MIT thesis, and how her work has blossomed from then to now.

**Dusa McDuff**

*Pseudoholomorphic curves in symplectic geometry*

**Raquel Mallavibarrena**

*Osculation for conic fibrations*

Abstract: Smooth projective surfaces fibered in conics over a smooth curve are interesting objects from the point of view of their k-th osculatory behavior, their structure plays a significant role for k ≥ 3. In the talk I will report about the main results and ideas of the joint research with Antonio Lanteri appearing in JPPA, 220, (2016). In particular, the possible existence of curves of low degree transverse to the fibers will be helpful to study the dimension of the osculation spaces of these surfaces. As an application we have obtained a complete description of the osculatory behavior of Castelnuovo surfaces and we have also studied the case k = 3 for the del Pezzo surfaces (the case k = 2 was already studied in another joint paper published in 2011). When the k-th inflectional locus has the expected codimension, a precise description of this locus is provided in terms of Chern classes. The final part of the talk will be devoted to some research in progress about projective bundles enveloping rational conic fibrations. This is also joint work with Antonio Lanteri.

**Bernard Teissier**

*Local polar varieties, geometry and algebra*

Abstract: A presentation of some of the relations between the multiplicity and the behavior of local polar varieties and the local geometry of a complex analytic set.

**Alicia Dickenstein**

*Higher order selfdual toric varieties*

The notion of higher order dual varieties of a projective variety is a natural generalization of the classical notion of projective duality, introduced by Ragni back in 1983. In this joint work with her, we give different combinatorial, algebraic and computational characterizations of those equivariant projective toric embeddings that satisfy higher order selfduality.

**Israel Vainsencher**

*Geometry of the Spaces of Holomorphic Foliations in \(\mathbb{CP}^n\) *

Abstract: Holomorphic foliations are a ``modern'' geometrical incarnation of systems of differential equations with polynomial coefficients. The Zariski closure of the set of foliations defined by a differential 1-form of type aFdG-bGdF, where F,G denote co-prime homogeneous polynomials of degrees a, b is an irreducible component of the space of foliations of codimension one and degree a+b-2. We give a formula for the degree of this component for a=2, b odd.

**Nelly Villamizar **

*Algebraic spline geometry*

Abstract: Splines are piecewise polynomial functions defined on a polyhedral partition of a real domain. These functions have many practical applications, including the finite element method for solving partial differential equations, geometric modeling, and computer graphics. From their definition, the study of spline functions involves an interplay between the underlying combinatorics and geometry of the partition and the algebraic properties of the resulting space of functions.

In this talk, we shall discuss the structure as a ring of the space of continuous spline functions defined on simplicial complexes embedded in a real space. It is well known that such a space of functions is a quotient of the Stanley-Reisner ring of the corresponding simplicial complex, and that the geometric realization of the Stanley-Reisner ring reflects the structure of the simplicial complex. We shall consider the generalized Stanley-Reisner rings associated to a simplicial complex, namely the ring of spline functions with higher order of global continuity on the simplicial complex. A conjectural description of the geometric realization of the generalized ring will be presented as part of our ongoing work with R. Piene on the topic.

**Elisa Postinghel**

*Tropical compactifications, Mori Dream Spaces and Minkowski bases*

Abstract: Given a Mori Dream Space X, we construct via tropicalisation a model dominating all the small Q-factorial modifications of X. Via this construction we recover a Minkowski basis for the Newton-Okounkov bodies of Cartier divisors on X and hence generators of the movable cone of X. This is joint work with Stefano Urbinati.

**Lothar Göttsche**

*Curve counting and refined curve counting.*

Abstract: The Severi degrees count the numbers of curves with a given number of nodes in a linear system on a surface. They have be studied for many years, also by Ragni Piene and Steve Kleiman. After reviewing some of the results I will explain a refinement of this count and relate it to real algebraic geometry and tropical geometry.

**Sandra Di Rocco**

*Higher order Gauss maps*

Abstract: The Gauss map is an important and fundamental tool in algebraic and differential geometry. When the complex projective variety is smooth, non degenerate and not linear the Gauss map is birational. We will present generalisations, so called higher order Gauss maps, capturing higher order local positivity and tangency. The main goal of the talk is to explain these (rational) maps and to show a straight forward generalisation of the finiteness property in the classical case. If time permits a toric interpretation will be presented. This is joint work with A. Lundman and K. Jabbusch.

**Jørgen Vold Rennemo **

*The Torelli theorem for cubic 4-folds via derived categories*

Abstract: The Torelli theorem for cubic 4-folds states that a cubic 4-fold X is determined up to isomorphism by the Hodge structure on its middle primitive cohomology, and was first proved by C. Voisin in 1985. I'll present a new proof of this theorem, which passes through the derived category of coherent sheaves of X. The starting point is Kuznetsov's observation that the derived category of X contains a full subcategory of "K3 type", which means that we can apply results relating Hodge theory and derived categories for K3 surfaces. This is joint work with D. Huybrechts.

**Paolo Aluffi**

*Chern classes of Schubert varieties*

Abstract: We compute the Chern-Schwartz-MacPherson classes of Schubert varieties in flag manifolds. These classes are obtained by constructing a representation of the Weil group by means of certain Demazure-Lusztig type operators. The construction extends to the equivariant setting. Based on explicit computations in low dimension, we conjecture that these classes are Schubert-positive; the analogous conjecture for Schubert varieties of Grassmannians was recently proven by June Huh. This is joint work with Leonardo Mihalcea.