Cayley-Bacharach condition and applications

Let X be a complex projective variety and let |D| be a complete linear system on X.

In line with the famous Cayley-Bacharach theorem, we say that a finite set S of points of X satisfies the Cayley-Bacharach condition with respect to |D| if any effective divisor of |D| passing through all but one point of S, passes also through the last point.

In this talk, we report on a joint work with Nicola Picoco, and we discuss the Cayley-Bacharach condition on projective spaces and Grassmannians, along with some applications.

In particular, we describe how Cayley-Bacharach condition on the Grassmannian G(k,n) imposes restrictions on the linear span of the corresponding k-planes in the n-dimensional projective space.

Moreover, we apply our result to study the covering gonality of the k-fold symmetric product of a smooth curve C. Namely, we prove that if k=2,3,4, the covering gonality of the k-fold symmetric product equals the gonality of C, and we characterize families of curves computing this invariant when C is a general curve.