Gröbner deformations
In this survey the author collects some recent results about connectedness and cohomological properties of the zero loci \(\mathcal Z(I)\subseteq\mathbb C\mathrm P^n\) of homogeneous ideals \(I\) in \(\mathbb C[x_0,x_1,\ldots,x_n]\). These results are mostly concerned with comparing properties and invariants associated to a homogeneous ideal \(I\) and to its initial ideal \(\mathrm{in}(I)\) (the ideal in \(\mathbb C[x_0,x_1,\ldots,x_n]\) generated by the leading monomials of the polynomials in \(I\) w.r.t.\ a fixed monomial ordering). For example, a theorem (which is proved in [\textit{M. Varbaro}, J. Algebra 322, 2492--2507 (2009; Zbl 1200.13029)]) states that if \(\mathcal Z(I)\) is connected, then \(\mathcal Z(\mathrm{in}(I))\) is also connected, and the converse is true if \(\mathrm{in}(I)\) is a radical ideal. In the last part of the survey, the author consideres projective schemes in \(\mathbb C\mathrm \mathbb P^n\) and discusses some related results and conjectures in terms of their cohomological invariants.
scientific article; zbMATH DE number 7342987