Seshadri Constant¶
A local positivity invariant of a nef line bundle at a point, computed by the least ratio of curve intersection to curve multiplicity, equivalently by the nef threshold on the blowup.
Core Idea¶
The Seshadri constant \(\varepsilon(L;x)\) measures how much local positivity a nef line bundle \(L\) has at a point \(x\) of a projective variety \(X\). For an irreducible curve \(C\subset X\) through \(x\), the intersection number \(L\cdot C\) measures global degree along \(C\), while \(\operatorname{mult}_x C\) measures how heavily that curve concentrates at \(x\). Their ratio penalizes curves that consume much multiplicity with little degree:
Scope of Application¶
Seshadri constants quantify local positivity of line bundles on projective varieties. They enter criteria for ampleness, lower bounds for separation of jets, behavior of adjoint linear systems, and comparisons between general and special points. Ein, Küchle, and Lazarsfeld prove uniform lower bounds at very general points in fixed dimension, while emphasizing that bounds valid at every point cannot ignore the particular variety and bundle.
Clarity¶
For \(X=\mathbf P^n\), \(L=\mathcal O_{\mathbf P^n}(1)\), and any point \(x\), a line through \(x\) gives ratio \(1/1=1\), so \(\varepsilon(L;x)\le1\). Degree/multiplicity bounds give \(\deg C\ge\operatorname{mult}_xC\), hence every curve ratio is at least \(1\). Therefore \(\varepsilon(\mathcal O(1);x)=1\).
Manages Complexity¶
Local positivity involves all curves through one point, each with global degree and local singularity behavior. The constant compresses this family into one extremal number without discarding the geometric obstruction: a curve approaching the infimum identifies where positivity is weakest. The blowup form reorganizes the same complexity into convex geometry of divisor classes.
Abstract Reasoning¶
The invariant is a constrained worst-case ratio. It balances a resource \(L\cdot C\) against a local demand \(\operatorname{mult}_xC\) and asks which curve produces the least margin. The blowup reveals a dual view: increase the local penalty \(tE\) until the global nonnegativity constraint becomes active.
Knowledge Transfer¶
The structural idea transfers to several marked points by blowing them up and subtracting a weighted sum of exceptional divisors. It also motivates moving Seshadri constants that discount fixed base loci. Such transfer is valid only after redefining the admissible curve family and normalization; a multipoint value cannot be silently substituted for \(\varepsilon(L;x)\).
Analogies to condition numbers or local Lipschitz constants can aid intuition because each records worst-case local behavior, but they are not aliases.
Relationships to Other Abstractions¶
Current abstraction Seshadri Constant Domain-specific
Parents (1) — more general patterns this builds on
-
Seshadri Constant presupposes Optimization Prime
Seshadri Constant compositionally presupposes Optimization: its definition specifies an admissible family of curves and takes an infimum, while its blowup form takes a supremal feasible coefficient.
Hierarchy path (1) — routes to 1 parentless root
- Seshadri Constant → Optimization
Neighborhood in Abstraction Space¶
Seshadri Constant sits in a sparse region of the domain-specific corpus (90th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Minimal Model Program — 0.82
- Castelnuovo–Mumford Regularity — 0.81
- Segre Class — 0.80
- Euler sequence — 0.78
- Circular Points at Infinity — 0.77
Computed from structural-signature embeddings · 2026-09-08