The classification of 4-dimensional empty lattice simplices.


Francisco Santos

University of Cantabria

Northeastern University

Thursday, September 21, 2017


Talk at 4:30 p.m. in 509 Lake Hall

Tea at 4:00 p.m. in 553 Lake Hall


Abstract: The classification of 4-dimensional empty lattice simplices was started in 1988 by Mori, Morrison, and Morrison (in what follows, MMM). Their motivation came from algebraic geometry where empty simplices correspond (almost) bijectively to terminal quotient singularities. More precisely:

- They identified 29 + 1 + 1 infinite families of them depending on one, two and three integer parameters respectively.

- They computationally enumerated all empty 4-simplices of prime determinant up to 1600 and found that beyond D=499 all of them belonged to these particular families.

They dubbed their conjecture that this behavior continued beyond their experimental window the "4-dimensional terminal lemma", since the usual "terminal lemma" was essentially equivalent to the classification of 3-dimensional empty simplices (which was first completed by White in 1964).

In this talk I will report on an extension of this classification to non-prime determinant. In particular, we found that:

- With non-prime determinant there is an additional two-parameter family and 23 additional one-parameter families to be added to the list by MMM.

- All simplices not belonging to these families again have determinant bounded above by 499.

Along the way, we correct a mistake made by Barile, Bernardi, Borisov and Kantor (2011) who wrongly assumed the MMM classification to hold also for non-prime volumes, and some minor errors in the tables given by MMM for the finitely many exceptional 4-simplices.

This is joint work with Oscar Iglesias-Valino. The talk will (try to) be non-technical and focus on what is known about classification of lattice simplices and polytopes in arbitrary dimensions, and on the ideas behind the proofs.

Here are some directions to Northeastern University. Lake Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street.

There is free parking available for people coming to the Colloquium at Northeastern's visitor parking (Rennaisance Garage). The entrance is from Columbus Avenue. If coming by car, you should park there and take the parking talon. After the lecture, you may pick up the payment coupon from one of NEU colloquium organizers.

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Posted: August 13, 2017    URL: