Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.

Determining the rank of a tensor has always been an interesting and important problem in algebraic complexity theory[8], algebraic statistics[3, 9], engineering[4] and algebraic geometry[6]. In this paper, I will first give a criterion for a generic m × n × n to have rank n. Right after the criterion, the first application is given. Then the symmetric version of this criterion is formulated. In Section 4, I will give a detailed discussion of the (O(1, 2) symmetric) ranks of (O(1, 2) symmetric) 3 × 2 × 2 tensors over the complex and real numbers with the aid of these criterions. This criterion also provides another way to attack the ”Salmon Problem ” over real numbers. 1. The Criterion In this section, we are working over any fixed field K. For a m×n×n tensor X, let X1, X2, · · · , Xm (which are n × n matrices) denote the slices in the first direction. Theorem. Let X be a m × n × n tensor with X1 nonsingular. Then X has rank n if the set of matrices {XjX −1 1: j = 2,...m} can be diagonalized simultaneously. Remark. The condition of the theorem can be weakened. In fact, if there exists a nonsingular linear combination of slices X1, X2, · · ·, Xm, then we can just replace X1 by that linear combination. This operation doesn’t change the rank at all. Note also that all linear combinations of Xi are singular is an algebraic condition, it amounts to say that det ( ∑ n i=1 λiXi) ≡ 0 for all λi ∈ C, i.e. all the coefficients of λi are

The hypersurface of Lüroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley’s proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes, studied by Coble, associated to the Cremona hexahedral representations of a nonsingular cubic surface. We also discuss the invariant-theoretical significance of these results and we give some new information about the locus of singular Lüroth quartics.

Abstract. In this paper we introduce a numerical invariant which imposes obstructions for smoothability of certain 0-dimensional schemes. This allows for the construction of families of nonsmoothable 0-schemes. Conversely, in low degree, the vanishing of these obstructions is sufficient for smoothability. We use the tools introduced in this paper to characterize nonsmoothable 0-schemes of minimal degree in every embedding dimension d ≥ 4, and to provide other results about Hilbert schemes of points. 1.