Squarefree Gröbner degenerations, special varieties and related topics

Commutative algebra is living an exciting moment. Classical tantalizing problems like the Eisenbud-Goto, Stillman’s, the Direct Summand and the Total Rank conjectures were recently settled: negatively in the first case, positively in the others.

The main focus of this project is on topics from commutative algebra, but a significant part of them have a strong overlap with algebraic geometry. The proposal revolves around new techniques provided by Conca and Varbaro’s recent solution of a conjecture of Herzog concerning squarefree Gröbner degenerations.

The proposal is divided into eleven subprojects, which are labeled as follows:

A. SMOOTH VARIETIES WITH A SQUARE-FREE GRÖBNER DEGENERATION

B. RATIONALITY VS UNIRATIONALITY OF SMOOTH HYPERSURFACES OF SMALL DEGREE WITH SPECIAL REGARD TO THOSE OF DIMENSION 3 AND 4

C. BLOWUP ALGEBRAS OF DETERMINANTAL IDEALS

D. DEFORMATION OF F-INJECTIVITY

E. REGULARITY AND LINEAR SYZYGIES

F. BARTH-IONESCU CONJECTURE FOR COMPLETE INTERSECTIONS

G. HILBERT SCHEMES OF LOCALLY COHEN-MACAULAY CURVES H. SINGULARITIES OF HILBERT SCHEMES

I. SPECIAL CLASSES OF IDEALS ADMITTING SQUARE-FREE MONOMIAL INITIAL IDEALS L. QUADRATIC QUOTIENTS OF REES ALGEBRAS

Some of these subprojects are also subtly connected to more distant areas of mathematics; for instance, issues from number theory on ordinary prime numbers of smooth projective varieties, or hyperbolicity issues in geometric group theory arise.

Several of the above topics, many of which seem unrelated, interplay with each other in a very promising wayThis is possible thanks to a strong synergy already present among the three research units of Catania, Genova and Milano.

Several topics we will deal with are tightly connected to “more applied” issues. For example, in the context of machine learning and signal processing, our project can contribute to tackle phase retrieval and computer vision issues. In another direction, there are strong relations with problems from post-quantum cryptography. Several researchers involved in this project have already demonstrated a strong ability to actively contribute in these directions.

Finally, our project is also very concerned with public engagement and dissemination of research results. Among other initiatives, we plan to produce a video about the use of Gröbner bases as a tool for solving polynomial systems and automatic deduction of simple geometric theorems, on the model of PBS Infinite Series and MSRI Numberphile.