VAT Seminar Schedule

Venues: (Tuesdays and Fridays) IMPAN 321/403 and (Thursdays) MIMUW 5050

The Organizers:
Stiofáin Fordham
Luke Oeding (contact: )
Emanuele Ventura

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Week XIV (Nov. 26–30)

Tuesday, Nov. 27

Secant varieties, tensor decomposition and identifiability

Speaker: Massimiliano Mella (Ferrara)
10:30–12:00, IMPAN 403

Abstract: The aim of these lectures is to introduce aspects of projective and birational geometry in the realm of tensor decompositions with particular regards to the identifiability problem.
After a brief review of secant varieties, with the help of Terracini's Lemma and the Infinitesimal Bertini Theorem, I will make a link between identifiability and weakly defective varieties. Then I will study the weakly defectiveness of Veronese varieties and as an application I will state, with a proof sketch, the complete classification of (generically) identifiable homogeneous polynomials. If time will allow I will discuss similar problems for arbitrary tensors.

Lecture I

Secant varieties, tensor decomposition and identifiability

Speaker: Massimiliano Mella (Ferrara)
14:15–15:15, IMPAN 321


Lecture II Abstract

The talk will be followed by tea and cake in 409 IMPAN at 15:15.

Thursday, Nov. 29

Fujita vanishing, sufficiently ample line bundles, and cactus varieties

Speaker: Jarosław Buczyński (IMPAN and MIMUW)
12:00–12:45 and 13:15–14:00, MIMUW 5050


For a fixed projective manifold $X$, we say that a property $P(L)$ (where $L$ is a line bundle on $X$) is satisfied by sufficiently ample line bundles if there exists a line bundle $M$ on $X$ such that $P(L)$ hold for any $L$ with ample $L-M$. I will discuss which properties of line bundles are satisfied by the sufficiently ample line bundles --- for example, can you figure out before the talk, whether a sufficiently ample line bundle must be very ample? A basic ingredient used to study this concept is Fujita's vanishing theorem, which is an analogue of Serre's vanishing for sufficiently ample line bundles. At the end of the talk I will define cactus varieties (an analogue of secant varieties) and sketch a proof that cactus varieties to sufficiently ample embeddings of $X$ are (set-theoretically) defined by minors of matrices with linear entries. This is closely related to conjectures of Eisenbud-Koh-Stillman (for curves) and Sidman-Smith (for any varieties). This is based on a joint work with Weronika Buczyńska and Łucja Farnik.

The talk will have a lunch break at the MIMUW Cafeteria.