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For any truncated path algebra $\Lambda$ of a quiver, we classify, by way ofrepresentation-theoretic invariants, the irreducible components of theparametrizing varieties $\mathbf{Rep}_{\mathbf{d}}(\Lambda)$ of the$\Lambda$-modules with fixed dimension vector $\mathbf{d}$. In this situation,the components of $\mathbf{Rep}_{\mathbf{d}}(\Lambda)$ are always among theclosures $\overline{\mathbf{Rep}\,\mathbb{S}}$, where $\mathbb{S}$ traces thesemisimple sequences with dimension vector $\mathbf{d}$, and hence the key tothe classification problem lies in a characterization of these closures.  Our first result concerning closures actually addresses arbitrary basicfinite dimensional algebras over an algebraically closed field. In the generalcase, it corners the closures $\overline{\mathbf{Rep}\,\mathbb{S}}$ by means ofmodule filtrations "governed by $\mathbb{S}$", in case $\Lambda$ is truncated,it pins down the $\overline{\mathbf{Rep}\,\mathbb{S}}$ completely.  The analysis of the varieties $\overline{\mathbf{Rep}\,\mathbb{S}}$ leads toa novel upper semicontinuous module invariant which provides an effective tooltowards the detection of components of $\mathbf{Rep}_{\mathbf{d}}(\Lambda)$ ingeneral. It detects all components when $\Lambda$ is truncated.

Closures in varieties of representations and irreducible components