Matrix factorizations and families of curves of genus 15 (Q2794758)

The purpose of this paper is to obtain results on the moduli space \(\mathcal M_{15}\) of curves of genus \(15\). By Brill-Noether theory, a general such curve has a smooth model of degree 16 in \(\mathbb P^4\), so the author focuses his interest on the (unique) component \(\mathcal H \subset \mathrm{Hilb}_{16t+1-15}(\mathbb P^4)\) of the Hilbert scheme of curves of degree \(d=16\) and genus \(g = 15\) in \(\mathbb P^4\) which dominates the moduli space \(\mathcal M_{15}\). It is not hard to see that a general element \(C \in \mathcal H\) lies on a unique smooth cubic hypersurface \(X \subset \mathbb P^4\) defined by a homogeneous polynomial, say \(f\), of degree 3. The notion of a matrix factorization of \(f\) was introduced by \textit{D. Eisenbud} [Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]; one produces a pair \((\phi,\psi)\) of matrices satisfying \(\psi \circ \phi =f \mathrm{id}\) and \(\phi \circ \psi = f \mathrm{id}\). From this one obtains a maximal Cohen-Macaulay module on the hypersurface ring \(R/f\). Setting \(\widetilde{\mathcal M}^4_{15,16} \) to be the component of \( \{(C,L) \mid C \in \mathcal M_{15}, L \in W_{16}^4 (C) \}\) which dominates \(\mathcal M_{15}\), the author proves that \(\widetilde{\mathcal M}^4_{15,16} \) is birational to a component of the moduli space of matrix factorizations of type \((\psi:\mathcal O^{18}(-3) \rightarrow \mathcal O^{15}(-1) \oplus \mathcal O^3(-2), \phi:\mathcal O^{15}(-1) \oplus \mathcal O^3(-2) \rightarrow \mathcal O^{18})\) of cubic forms on \(\mathbb P^4\). As a corollary he shows that a general cubic threefold in \(\mathbb P^4\) contains a 32-dimensional uniruled family of smooth curves of genus 15 and degree 16. He also shows that the moduli space \(\widetilde{\mathcal M}^4_{15,16} \) is uniruled, and produces a probabilistic algorithm to randomly produce curves of genus 15 over a finite field \(\mathbb F_q\) with \(q\) elements from a Zariski open subset of \(\mathcal M_{15}\) in running time \(O((\log q)^3)\). The original goal was to prove that \(\mathcal M_{15}\) is unirational, and he ends with a conjecture about the obstruction. He used the software package Macaulay2 in this work, and an important tool was Boij-Söderberg theory for a list of candidate Betti tables in the argument.