Lifting negative curves from finite fields and Seshadri constants at ten points

Abstract

We use finite-field calculations and mixed-characteristic deformation to construct nef divisors on very general complex blow-ups of the plane. A lifting and smoothing criterion converts a reducible nodal divisor over ${\mathbb{F}}_{127}$ into a smooth genus-$10$ curve of class $117H-37(E_1+\cdots +E_{10})$, retaining control of its first-order obstruction map. Its normal bundle in a suitable deformation is semistable, so Petrakiev's multiplicity estimate yields the nef class $27379H-8658(E_1+\cdots +E_{10})$ at ten very general points. Their multipoint Seshadri constant is therefore at least $8658/27379$, improving Petrakiev's bound $228/721$. The geometric input is an existence criterion in characteristic zero verified by computations in positive characteristic.

12 pages