Proper-point defects and the degree-21 bottleneck

Definitions and mathematical statement

For a singular reduced plane curve $C$ of degree $d$, let $x_1,\ldots,x_s$ be its distinct proper singular points and $m_i$ their multiplicities, with $s>0$. The supplied source convention is \[ h(C)=\frac{d^2-\sum_{i=1}^s m_i^2}{s}. \] Only proper points in $\mathbb P^2$ occur in this sum and denominator; infinitely near points are not additional counted singular points.

AIM problem 5.3. For irreducible curves, the source reports the best known examples approaching $h=-2$ and asks whether one can do better. Seek irreducible examples with smaller values or an improved asymptotic family, specifying which improvement is meant. A single value below $-2$ and a limiting family below $-2$ are different targets; the excerpt does not formalize that distinction or explicitly state the base field.

Original Remark 5.4 credits Roulleau with configurations of cubics having $h\to-4$; configurations need not satisfy the irreducibility requirement. Remark 5.5 reports $h\to-3$ and $h\ge-3$ over ${\mathbb{R}}$, no analogous result known over ${\mathbb{Q}}$ at the time, and asks about bounded negativity for tropical surfaces. The exact curve/configuration class for the real bound and the tropical intersection convention are not restated, so these are retained as qualified historical remarks rather than unrestricted theorems.

Supplied research-aid context, not a verified solution. For a reduced irreducible complex plane curve, let $g$ be the genus of its normalization and $\delta_i$ the local delta invariants. The attempt defines \[ \eta_i=2\delta_i-m_i(m_i-1) \] and claims these are even nonnegative infinitely-near defects. It reports the exact identities \[ s(h+2)=3d-2+2g-\sum_i(m_i-2)+\sum_i\eta_i, \] \[ 3s(h+2)=\Lambda+7g+\frac{-d^2+21d-14}{2}, \] where $\Lambda$ is described as a sum of nonnegative local slacks but is not explicitly defined in the record.

The report claims $h>-2$ in degrees at most twenty. For degree twenty-one with $h<-2$, it gives necessary conditions \[ g=0,\qquad\eta_i=0\text{ for every }i,\qquad \Lambda\in\{1,4\}, \] and refers to a short explicit list of multiplicity profiles, which is not supplied. It concludes that $h$ must be $-2-2/s$ or $-2-1/s$. Vanishing of $\eta_i$ is the stated numerical defect condition and is not silently identified with every singularity being ordinary. These are proposed necessary-pattern restrictions, not existence of any degree-twenty-one curve or an answer to the example-construction problem.

Short English statement

Find irreducible plane curves improving the source's $h\to-2$ examples, using its proper-singular-point convention. The attached degree-twenty-one restrictions identify only possible patterns and do not construct such curves.

Sources

• Local UnsolvedMath registry, permanent ID 20000225, source code catalogue entry; no proposer or proposal year supplied.
• AIM workshop Degenerations in Algebraic Geometry, six-page problem-session record, September 7, 2016, Problem 5.3 and Remarks 5.4--5.5; supplied locator https://aimath.org/pastworkshops/degenalggeomproblems.pdf. Canonical locator: aim-algebraic-geometry-notes.json, notes[224], tag "problem." Original literature field is empty. No PDF was fetched or generated here.
• Roulleau is named in the original remark without a full citation. Research attempt 1: partial_result, status at run "open," machine-assigned rigor "proved," novelty confidence "low," with an explicit machine-generated/not-peer-reviewed/verify-before-citing caveat.