Abstract. We classify the bipartite graphs G whose binomial edge ideal JG is Cohen-Macaulay. The connected components of such graphs can be obtained by gluing a finite number of basic blocks with two operations. In this context we prove the converse of a well-known result due to Hartshorne, showing that the Cohen-Macaulayness of these ideals is equivalent to the connectedness of their dual graphs. We study interesting properties also for non-bipartite graphs and in the unmixed case, constructing classes of bipartite graphs with JG unmixed and not Cohen-Macaulay.

In this paper we introduce a natural model for the realization space of a polytope up to projective equivalence which we call the slack realization space of the polytope. The model arises from the positive part of an algebraic variety determined by the slack ideal of the polytope. This is a saturated determinantal ideal that encodes the combinatorics of the polytope. We also derive a new model of the realization space of a polytope from the positive part of the variety of a related ideal. The slack ideal offers an effective computational framework for several classical questions about polytopes such as rational realizability, non-prescribability of faces, and realizability of combinatorial polytopes.Lemma 2.1 ([GGK + 13, Theorem 14]). If S is any slack matrix of P , then the polytope Q = conv(rows(S)), is affinely equivalent to P .By the above discussion, we may translate P so that 0 ∈ int(P ) without changing its slack matrices. Subsequently, we may scale facet inequalities to set w = ½. Then

In this paper, we study orthogonal representations of simple graphs G in R d from an algebraic perspective in case d = 2. Orthogonal representations of graphs, introduced by Lovász, are maps from the vertex set to R d where nonadjacent vertices are sent to orthogonal vectors. We exhibit algebraic properties of the ideal generated by the equations expressing this condition and deduce geometric properties of the variety of orthogonal embeddings for d = 2 and R replaced by an arbitrary field. In particular, we classify when the ideal is radical and provide a reduced primary decomposition if √ −1 ∈ K. This leads to a description of the variety of orthogonal embeddings as a union of varieties defined by prime ideals. In particular, this applies to the motivating case K = R.2010 Mathematics Subject Classification. 05E40, 13C15, 05C62, 05E99.Proposition 2.3. The ideals in < (I Kn ) and in < (I K m,n−m ) have a linear resolution, height(in < (I Kn )) = n, height(in < (I K m,n−m )) = n − 1, depth(S/ in < (I Kn )) = 1 and S/ in < (I K m,n−m ) is Cohen-Macaulay. The same statements hold for the ideals I Kn and I K m,n−m .Proof. We first show that in < (I Kn ) has a linear resolution. By Lemma Lemma 2.1,We order the generators of this initial ideal in a way that the monomial generators of J 1 are bigger than the monomial generators of J 2 and such that the generators of J 1 as well as the generators of J 2 are ordered lexicographically induced by x 1 > · · · > x n > y 1 > · · · > y n . The ideal J has linear quotients with respect to this ordering of its monomial generators. Indeed, the ideal J 1 is known to have linear quotients. Now let x i y j ∈ J 2 . We denote by J ij the ideal generated by all monomial generators of J which are bigger than x i y j . Then J ij : x i y j = (x 1 , . . . , x n , y i+1 , . . . , y j−1 ). This shows that J has linear quotients and hence has a linear resolution by [4, Proposition 8.2.1]. Moreover it follows from [4, Corollary 8.2.2] that proj dim(J) = 2n − 2, because J 1n

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.

A. We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated in any degree, proving that the same equality holds. (2010): 13A15, 13F55, 14M10, 05C05, 05C38. Mathematics Subject ClassificationGiven a Noetherian commutative ring with identity R, the arithmetical rank (ara) of a proper ideal I of R is defined as the smallest integer s for which there exist s elements a 1 , . . . , a s of R such that the ideal (a 1 , . . . , a s ) has the same radical as I. In this case we will say that a 1 , . . . , a s generate I up to radical. In general ht I ≤ ara I. If equality holds, I is called a set-theoretic complete intersection. As a consequence of the Auslander-Buchsbaum formula, whenever an ideal of R = k[x 1 , . . . , x n ] generated by squarefree monomials is a set-theoretic complete intersection, it is a Cohen-Macaulay ideal. The converse is not always true. We consider the case where R is a polynomial ring over a field K and I is the so-called edge ideal of a graph whose vertices are the indeterminates. Its set of generators is formed by the products of the pairs of indeterminates that form the edges of the graph. Thus I is generated by squarefree monomials of degree 2, and is therefore a radical ideal. Large classes of graphs whose edge ideals are Cohen-Macaulay were described by Villarreal [16]. The arithmetical rank of edge ideals has recently been studied by several authors (see e.g. Kummini [12]) and explicitly determined for some special types of graphs. According to a well-known result by Lyubeznik [13], if I is a squarefree monomial ideal, the projective dimension of the quotient ring R/I, denoted pd R R/I, provides a lower bound for the arithmetical rank of I. We define the big height of I, denoted bight I, as the maximum height of the minimal prime ideals of I. In general, we have ht I ≤ bight I ≤ pd R R/I ≤ ara I. If I is not unmixed, then I is not a set-theoretic complete intersection, but it could still be true that bight I = pd R R/I = ara I. This equality has been established for the edge ideals of acyclic graphs (the so-called forests) by Kimura and Terai [11] (extending a result by Barile [1]). A weaker condition is the equality between the arithmetical rank and the projective dimension. This is the case for lexsegment edge ideals (see Ene, Olteanu, Terai [6]), for the graphs formed by one or two cycles connected through a path (cyclic and bicyclic graphs, see Barile, Kiani, Mohammadi and Yassemi [2]) and for the graphs consisting of paths and cycles with a common vertex (see Kiani and Mohammadi [9]). In all these cases, the arithmetical rank is independent of the field K. As a consequence of what we said above, the classes of Cohen-Macaulay monomial ideals are candidate to be set-theoretic complete intersections. We consider the family of whisker graphs, obtained by adding a whisker to each vertex of a given graph, i.e....