The resultant divisor is negative

TL;DR

The paper studies the birational geometry of the parameter space $P_{d,e}$ for pairs of homogeneous polynomials of degrees $d<e$ in two variables, proving that the minimal model program can be run on this space and that it contracts the resultant divisor $\Delta_{d,e}$, thereby producing a projective compactification of the open locus $U_{d,e}$ without the discriminant. Central to the approach is the construction of a smooth model $\widehat{P}_{d,e}$ via successive blow-ups, its identification with the multigraded Hilbert scheme $\mathrm{Hilb}_{d,e}$, and the realization of a family of birational models through semi-ample line bundles $\mathcal{L}_{d,e}^i$. The work establishes that $P_{d,e}$ is a Mori dream space with an effective cone generated by $\mathcal{O}(1,0)$ and $\mathcal{O}(\Delta_{d,e})$, and provides a detailed MMP diagram whose last model is a codimension-two boundary compactification of $U_{d,e}$ with strata mirroring $(U_{d-j,e+j})$. In addition to geometric results, the paper connects to invariant theory by showing the Cox ring of $P_{d,e}$ corresponds to a finitely generated invariant ring $\mathcal{O}(V_d\times V_e)^{V_{e-d}}$, illustrating how the MMP informs the structure of invariants and Hilbert schemes. Overall, the results yield concrete birational models, construct projective curves avoiding the resultant divisor, and illuminate links between Mori theory, multigraded Hilbert schemes, and invariant theory.

Abstract

Fix two integers $1\leq d<e$. We study the birational geometry of a parameter space for pairs of homogeneous polynomials of degrees $d$ and $e$ in two variables (in which the higher degree polynomial is well defined only up to a multiple of the lower degree polynomial). We show that one can run the MMP on this space, and that it eventually contracts the resultant divisor.

Theorems & Definitions (60)

• Theorem 1.1: Corollary \ref{['corcompactification']}
• Theorem 1.2: Theorem \ref{['completecurves']}
• Theorem 1.3: Theorem \ref{['thMMPmain']}
• Theorem 1.4: Theorem \ref{['thmCox']}
• Theorem 3.1
• Proposition 3.2
• Proposition 3.3
• proof
• Lemma 3.4
• proof