Mixed multiplicities of equigenerated ideals

Time: 9:30 - 11:00, 17th July

Venue: Room 507, A6, Institute of Mathematics

Abstract: Let $R=\oplus_{m\geq 0}R_m$ be a $d$-dimensional standard graded Noetherian domain over a field $R_0=k$ and $\mathfrak{m}$ be the unique graded maximal ideal of $R$. Let $I$ be a nonzero non $\mathfrak{m}$-primary graded ideal in $R$ generated in equal degrees. In this setup we shall study the mixed multiplicities $\{e_i(\mathfrak{m}\vert I)\}_{i=0}^{d-1}$, as defined by Katz and Verma. Our key observation is that the mixed multiplicities $e_i(\mathfrak{m}\vert I)$ can be realized as intersection numbers of nef divisors on a projective variety. As a consequence we recover a result of Huh about log-concavity of these mixed multiplicities. We also obtain other inequalities involving these invariants. Furthermore, we show that the last mixed multiplicity $e_{d-1}(\mathfrak{m}\vert \_)$ can be used to produce a Rees-type criterion for detecting integral closures of equigenerated ideals. If time permits, we shall also discuss how some of the results can be generalized to mixed multiplicities of filtrations of equigenerated ideals. This talk will be based on two ongoing projects: (1) with Roy and Trivedi, and (2) with Roy and Truong.