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Register a new userThe Minimal Resolution Conjecture on a general quartic surface in $mathbb P^3$
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Mustac{t}u{a} has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in $mathbb P^3$ this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.
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We provide a number of new conjectures and questions concerning the syzygies of $mathbb{P}^1times mathbb{P}^1$. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of $mathbb{P}^1times mathbb{P}^1$. These computations utilize linear algebra over finite fields and high-performance computing.
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Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R, )$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $ $-stable filtrations ${mathbb M} $ of $M $ and to compare the Betti numbers of $M$ with those of the associated graded module $ gr_{mathbb M}(M). $ This approach has the advantage that the same module $M$ can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Among others, the Koszul modules have this behavior. As a consequence of the main result, we extend some results by Aramova, Conca, Herzog and Hibi on the rigidity of the resolution of standard graded algebras to the local setting.
Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(bf b)$ for a degree $mathbf{b}in A$. We show that the homology of $D_{gcd}(bf b)$ determines the $i$-Betti numbers of degree $bf b$. We discuss the notion of an indispensable complex of $R/I$. We show that the Koszul complex of a complete intersection lattice ideal $I$ is the indispensable resolution of $R/I$ when the $A$-degrees of the elements of the generating $R$-sequence are incomparable.
Let $Z$ be a finite set of $s$ points in the projective space $mathbb{P}^n$ over an algebraically closed field $F$. For each positive integer $m$, let $alpha(mZ)$ denote the smallest degree of nonzero homogeneous polynomials in $F[x_0,ldots,x_n]$ that vanish to order at least $m$ at every point of $Z$. The Waldschmidt constant $widehat{alpha}(Z)$ of $Z$ is defined by the limit [ widehat{alpha}(Z)=lim_{m to infty}frac{alpha(mZ)}{m}. ] Demailly conjectured that [ widehat{alpha}(Z)geqfrac{alpha(mZ)+n-1}{m+n-1}. ] Recently, Malara, Szemberg, and Szpond established Demaillys conjecture when $Z$ is very general and [ lfloorsqrt[n]{s}rfloor-2geq m-1. ] Here we improve their result and show that Demaillys conjecture holds if $Z$ is very general and [ lfloorsqrt[n]{s}rfloor-2ge frac{2varepsilon}{n-1}(m-1), ] where $0le varepsilon<1$ is the fractional part of $sqrt[n]{s}$. In particular, for $s$ very general points where $sqrt[n]{s}inmathbb{N}$ (namely $varepsilon=0$), Demaillys conjecture holds for all $minmathbb{N}$. We also show that Demaillys conjecture holds if $Z$ is very general and [ sgemax{n+7,2^n}, ] assuming the Nagata-Iarrobino conjecture $widehat{alpha}(Z)gesqrt[n]{s}$.
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