Rational curves and instantons on the Fano threefold $Y_5$

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-Mülich theorem for the splitting type of instantons on conics, a bound to the splitting type of instantons on lines and an $S{L}_2$-equivariant description of the moduli space in charge 2 and 3. Using these results we prove the existence of a unique $S{L}_2$-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.

118 pages, PhD thesis

References in corpus (2)

• Homological projective duality for Grassmannians of lines
• Hochschild homology and semiorthogonal decompositions

Cited by in corpus (10)

• Hilbert schemes of lines and conics and automorphism groups of Fano threefolds
• Geometric Manin's Conjecture and rational curves
• Compactification of the moduli space of instanton sheaves on the Fano threefold $V_5$
• A desingularization of Kontsevich's compactification of twisted cubics in $V_5$
• Bridgeland stability of minimal instanton bundles on Fano threefolds
• Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture
• Rational curves on $V_5$ and rational simple connectedness
• Torsion free instanton sheaves on the blow-up of ${\mathbb{P}}^3$ at a point
• Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V_5 and V_22
• Okamoto Transformations relating Equivariant Instanton Bundles via Painleve VI