We construct wonderful compactifications of the spaces of linear maps and symmetric linear maps of a given rank as blowups of secant varieties of Segre and Veronese varieties. Furthermore, we investigate their birational geometry and their relations with some spaces of degree two stable maps.

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We construct wonderful compactifications of the spaces of linear maps, and symmetric linear maps of a given rank as blow-ups of secant varieties of Segre and Veronese varieties. Furthemore, we investigate their birational geometry and their relations with some spaces of degree two stable maps.

A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians. Contents 1. Introduction 1 2. Complete quadrics 4 3. Complete symmetric symplectic forms 8 4. Divisors on S 2r 14 5. Birational geometry of S 2r 17 6. Moduli spaces of conics in Lagrangian Grassmannians 20 References 28