Effective reconstruction of generic genus 5 curves from their theta hyperplanes

Effective reconstruction formulas of a curve from its theta hyperplanes are known since the ancients in genus 2 (where the theta hyperplanes are Weierstrass points), and -- assuming the theta hyperplanes are ordered -- in genus 3. A non-effective proof that the set of (non-ordered) odd theta characteristic of a generic general genus curve determine the curve was give in CS2003. In Le2015 we gave a partial solution to the effective problem in genus 4 (later completed by CKRN2019). In this work we extend from genus 4 to 5 the first part of Le2015: the reconstruction of the set of enveloping quadrics of a generic curve C from its theta hyperplanes; for a generic genus 5 curve C, this data suffices to reconstruct C. As a consequence we get a complete description of the Schottky locus in genus 5 in terms of theta hyperplanes. The computational part of the proof is a certified numerical argument.Comment: 23 pages + accompanying cpp computer program in the ancillary file

Lagrangian Fibrations by Prym Varieties

We describe and study Lagrangian fibrations including non-compact ones from Hitchin systems and compact ones from the moduli spaces of semi-stable sheaves of some fixed invariant on K3 surfaces. We prove properties of the relative Prym variety P constructed from a double cover of a degree one del Pezzo surface T by a K3 surface S using the bi-anticanonical class of T. P is a 6 dimensional singular holomorphic symplectic variety with generic fibers being Prym varieties of a new polarization type (1, 2, 2). We examine the singularities of P and show that P is birational to a quotient of a smooth simply-connected projective variety by an involution. In the end, we prove a degeneration from the relative Prym variety associated to del Pezzo surfaces to a natural compactification of the Sp(2n, C)-Hitchin system.Doctor of Philosoph

Tritangents and their space sextics

Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta characteristics. In this paper we revisit both results from the computational perspective. Specifically, we give an algorithm to construct space sextic curves that arise from blowing up projective plane at eight points and provide algorithms to compute the 120 tritangents and their Steiner system of any space sextic. Furthermore, we develop efficient inverses to the aforementioned methods. We present an algorithm to either reconstruct the original eight points in the projective plane from a space sextic or certify that this is not possible. Moreover, we extend a construction of Lehavi which recovers a space sextic from its tritangents and Steiner system. All algorithms in this paper have been implemented in magma.Comment: 24 pages, 2 figure