English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Schenck and Seceleanu proved that if $R = k[x,y,z]$, where $k$ is an infinitefield, and $I$ is an ideal generated by any collection of powers of linearforms, then multiplication by a general linear form $L$ induces a homomorphismof maximal rank from any component of $R/I$ to the next. That is, $R/I$ has the{\em weak Lefschetz property}. Considering the more general {\em strongLefschetz question} of when $\times L^j$ has maximal rank for $j \geq 2$, wegive the first systematic study of this problem. We assume that the linearforms are general and that the powers are all the same, i.e. that $I$ isgenerated by {\em uniform} powers of general linear forms. We prove that forany number of such generators, $\times L^2$ always has maximal rank. We thenspecialize to almost complete intersections, i.e. to four generators, and weshow that for $j = 3,4,5$ the behavior depends on the uniform exponent and on$j$, in a way that we make precise. In particular, there is always at most onedegree where $\times L^j$ fails maximal rank. Finally, we note thatexperimentally all higher powers of $L$ fail maximal rank in at least twodegrees.

On the strong Lefschetz question for uniform powers of general linear forms in $k[x,y,z]$