Grzegorz Kapustka, Jagiellonian University, Krakow, Poland. Michal Kapustka, Institute of Mathematics of the Polish Academy of Sciences, Warszawa, Poland. Kristian Ranestad, Universitetet i Oslo, Norway. Hal Schenck, Auburn University, Alabama, Mike Stillman, Cornell University, Ithaca, New York. Beihui Yuan, Beijing Institute of Mathematical Sciences and Applications, People's Republic of China.

• Chapters; 1. Introduction; 2. Preliminaries; 3. Apolar curves and Betti tables of point sets; 4. The quadratic part of the apolar ideal; 5. Rank and powersum presentations of quartics; 6. Stratifications of the space of quaternary quartics; 7. Codimension three varieties in quadrics; 8. Irreducible liftings; 9. Reducible liftings; A. Betti tables for the inverse system; B. Betti table subschemes of the Hilbert scheme

Grzegorz Kapustka, Jagiellonian University, Krakow, Poland. Michal Kapustka, Institute of Mathematics of the Polish Academy of Sciences, Warszawa, Poland. Kristian Ranestad, Universitetet i Oslo, Norway. Hal Schenck, Auburn University, Alabama, Mike Stillman, Cornell University, Ithaca, New York. Beihui Yuan, Beijing Institute of Mathematical Sciences and Applications, People's Republic of China.

• Chapters; 1. Introduction; 2. Preliminaries; 3. Apolar curves and Betti tables of point sets; 4. The quadratic part of the apolar ideal; 5. Rank and powersum presentations of quartics; 6. Stratifications of the space of quaternary quartics; 7. Codimension three varieties in quadrics; 8. Irreducible liftings; 9. Reducible liftings; A. Betti tables for the inverse system; B. Betti table subschemes of the Hilbert scheme

Grzegorz Kapustka, Jagiellonian University, Krakow, Poland. Michal Kapustka, Institute of Mathematics of the Polish Academy of Sciences, Warszawa, Poland. Kristian Ranestad, Universitetet i Oslo, Norway. Hal Schenck, Auburn University, Alabama, Mike Stillman, Cornell University, Ithaca, New York. Beihui Yuan, Beijing Institute of Mathematical Sciences and Applications, People's Republic of China.

• Chapters; 1. Introduction; 2. Preliminaries; 3. Apolar curves and Betti tables of point sets; 4. The quadratic part of the apolar ideal; 5. Rank and powersum presentations of quartics; 6. Stratifications of the space of quaternary quartics; 7. Codimension three varieties in quadrics; 8. Irreducible liftings; 9. Reducible liftings; A. Betti tables for the inverse system; B. Betti table subschemes of the Hilbert scheme