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Let $V:f=0$ be a hypersurface of degree $d \geq 3$ in the complex projective space $\mathbb{P}^n$, $n \geq 3$, having only isolated singularities. Let $M(f)$ be the associated Jacobian algebra and $H: \ell=0$ be a hyperplane in $\mathbb{P}^n$ avoiding the singularities of $V$, but such that $V \cap H$ is singular. We related the Lefschetz type properties of the linear maps $\ell: M(f)_k \to M(f)_{k+1}$ induced by the multiplication by linear form $\ell$ to the singularities of the hyperplane section $V \cap H$. Similar results are obtained for the Jacobian module $N(f)$.

Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surface $Y$ is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms on $Y$ quotiented by the torsion-free part of the first integral homology of $Y$. Denote by $X$ the double cover of $Y$ given by orientation. The Jacobian of $Y$ is identified with the space of all degree zero holomorphic line bundles $L$ over $X$ with the property that $L$ is isomorphic to $σ^*\bar{L}$, where $σ$ is the involution of $X$.

We prove that the quotient of Jacobian of a curve whose genus is greater than or equal to 5 under the action of a finite group acting on the curve is never uniruled, and classify all curves of genus 3 and 4 whose quotients of Jacobian is uniruled.

We prove by means of the study of the infinitesimal variation of Hodge structure and a generalization of the classical Babbage-Enriques-Petri theorem that the Jacobian variety of a generic element of a $k$ codimensional subvariety of $\mathcal M_g$ is not isogenous to a distinct Jacobian if $g>3k+4$. We extend this result to $k=1, g\ge 5$ by using degeneration methods.

We introduce jacobian graphs, which are explicit families of regular graphs that are spectrally indistinguishable from random graphs, but whose local structure is very different from that of random graphs. The construction relies on the geometric properties of generalized jacobians of curves and on general equidistribution theorems for character sums over finite fields.

The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from its Jacobian ideal.

We introduce a new method in the attempt to prove the Jacobian conjecture. In the complex dimension 2 case, we apply this method to prove some new results related the Jacobian conjecture.

We show that the endomorphism ring of each cluster tilting object in a tubular cluster category is a finite dimensional Jacobian algebra which is tame of polynomial growth. Moreover, these Jacobian algebras are given by a quiver with a non-degenerate potential and mutation of cluster tilting objects is compatible with mutation of QPs.

The Jacobian algebras are introduced and their various properties are studied.

We show that the Jacobian conjecture of the two dimensional case is true.

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