Rational curves and instantons on the Fano threefold $Y_5$

TL;DR

This thesis investigates the moduli spaces of instanton bundles on the Fano threefold Y_5, providing new proofs and several original results on their properties, including bounds, classifications, and compactifications.

Contribution

It introduces new theorems on instanton splitting types, describes the moduli space in low charges, and constructs a compactification with a resolution for charge 3 instantons.

Findings

Existence of a unique minimal charge instanton with SL_2 symmetry.

Smooth divisor of jumping lines for charge 2 instantons.

Examples of reducible jumping line divisors in charge 3 instantons.

Abstract

This thesis is an investigation of the moduli spaces of instanton bundles on the Fano threefold $Y_5$ (a linear section of $\mathbb{G}r(2,5)$). It contains new proofs of classical facts about lines, conics and cubics on $Y_5$, and about linear sections of $Y_5$. The main original results are a Grauert-M\"ulich theorem for the splitting type of instantons on conics, a bound to the splitting type of instantons on lines and an $SL_2$-equivariant description of the moduli space in charge 2 and 3. Using these results we prove the existence of a unique $SL_2$-equivariant instanton of minimal charge and we show that for all instantons of charge 2 the divisor of jumping lines is smooth. In charge 3, we provide examples of instantons with reducible divisor of jumping lines. Finally, we construct a natural compactification for the moduli space of instantons of charge 3, together with a small resolution of singularities for it.